Enumeration Problemsand

Methods of Modular Homology

Basant Dutt Sharma

School of Computing, Information and Mathematical SciencesFaculty of Science and TechnologyThe University of the South Pacific

February 2006

A thesis submitted in the partial fulfillment of the requirements of the degree ofMaster of Science

c©Basant Dutt Sharma 2006

Acknowledgement

I would like to take this opportunity to thank all those who contributed to thesuccessful completion of this thesis.

My profound thanks go to my supervisor, Dr Valeriy Mnukhin, of the Depart-ment of Mathematics and Computing at the University of the South Pacific, forintroducing me to the indispensable method of Modular Homology. I would liketo express my deeply felt appreciation for his valuable suggestions, motivationsand counseling in every phase of this arduous but meritorious task.

I am also thankful to the thesis examiners, Prof. Vyacheslav Futorny (exter-nal examiner) and Dr. Dmitry Malinin (internal examiner) for there valuablecomments.

Special thanks are also due to the Head of the Mathematics department, Dr.Jito Vanualailai, for providing assistance whenever there was a need.

I am greatly indebted to my family, my daughter Shreya and wife Asmi for hermotivation and incessant support. I also wish to thank friends and colleagues fortheir words of advice.

This thesis is dedicated to the memory of my parents Mr. and Mrs Ram Duttand my son Aditya. I can only wish that they were here to see this achievement.

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Declaration of Originality

I hereby declare that the work presented in this thesis is, to the best of myknowledge and belief, original, except as acknowledged in the text. The materialin the thesis has not been submitted previously, either in whole or part, for adegree at this or any other institution.

· · · · · · · · · · · · · · · · · · · · · · · ·Basant Dutt SharmaFebruary, 2006

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Contents

1 Introduction 5

2 Groups and Group Actions 7

2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 Permutation Groups and Orbits . . . . . . . . . . . . . . . 12

2.1.5 Normal Subgroups and Simple Groups . . . . . . . . . . . 14

2.1.6 Group Homomorphisms . . . . . . . . . . . . . . . . . . . 16

2.2 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Enumeration of Orbits . . . . . . . . . . . . . . . . . . . . 21

2.3 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Finite Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Homology Theory 28

3.1 The Homological Sequence . . . . . . . . . . . . . . . . . . . . . . 28

3.2 The Simplex and the Simplicial Complex . . . . . . . . . . . . . . 30

3.3 Homology of Simplicial Complexes . . . . . . . . . . . . . . . . . 31

4 Modular Homology Theory 35

4.1 Generalized Homology . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Composition of Inclusion Maps . . . . . . . . . . . . . . . . . . . 37

3

4.3 Modular Homology of Simplicial Complexes . . . . . . . . . . . . 38

4.4 Modular Homology of a Simplex Σn . . . . . . . . . . . . . . . . 40

4.5 Group Actions on a Simplex . . . . . . . . . . . . . . . . . . . . . 42

4.6 The Sequence of Invariant Spaces . . . . . . . . . . . . . . . . . . 44

4.7 Improvement of Inequalities . . . . . . . . . . . . . . . . . . . . . 49

4.8 Group Actions on Modular Homology . . . . . . . . . . . . . . . . 52

4.9 Generalized Livingstone and Wagner Inequalities . . . . . . . . . 54

4.10 Modular Homology of q -Simplex . . . . . . . . . . . . . . . . . . 58

5 Numbers of k -Orbits of Permutation Groups 62

5.1 Conditions For Non-triviality . . . . . . . . . . . . . . . . . . . . 63

5.2 Inequalities for Numbers of Orbits . . . . . . . . . . . . . . . . . . 64

5.3 Some General Results . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 Numbers of k -Orbits of Linear groups 74

6.1 Conditions for Nontriviality . . . . . . . . . . . . . . . . . . . . . 74

6.2 Subgroups of the General Linear Groups . . . . . . . . . . . . . . 75

6.3 Classical Linear Groups . . . . . . . . . . . . . . . . . . . . . . . 105

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Chapter 1

Introduction

My thesis is about applications of methods of modular homology to enumerationproblems related with group actions. Enumeration problems are probably theoldest problems of combinatorics. The classical problem of finding the numberof orbits of a permutation group G that acts on subsets ( see [6]), has been ofparticular interest. Well-known solution of this problem is based on the Polya’sTheory and requires knowledge of the cycle index of the group G . Unfortunately,the cycle index is usually too complicated to derive any general results about theorbits. To derive such general results, different methods are needed.

The method based on investigation of so-called inclusion maps has been used byD. Livingstone and A. Wagner in [15] to prove the inequalities between numbersof orbits. It occurs that the inequalities of Livingstone and Wagner hold in moregeneral situations, for example, for numbers Nk , of orbits of linear groups thatact on k -dimensional subspaces. At the same time no method, similar to thePolya’s Theory, to determine the numbers Nk seems to be known. Some partialresults have been found in [36, 37].

Further investigations of inclusion maps have been done by T. Bier [3, 4], PeterFrankl [10], A. Frumkin and A. Yakir [11], V. Mnukhin and J. Siemons [17],R. Stanley [29], R. Wilson [34] and other mathematicians. It has been provedin [3, 17] that some special sequences of inclusion maps are homological andso the inclusion maps (over finite fields) could be used to build a generalizedmodular homology theory. Generalized homology appears to be mentioned firstby W. Mayer [16] in 1942 and then studied by Spanier [28]. More recent paperson generalized homology include C. Kassel and M. Wambst [13], M. Dubois-Violette [9], M. Kapranov [14], A. Tikaradze [33].

As it has been shown in [17, 21, 23], modular homology provides efficient tech-nique to study group actions both on subsets and on subspaces. The resultsof [23], improve the inequalities of Livingstone and Wagner and in some casescould be used to produce nontrivial results about numbers of orbits. One of the

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motivations for my Thesis has been, the systematic exploration of cases whenthe inequalities of Livingstone and Wagner could be improved.

The organization of my Thesis is as follows. Chapters 2 and 3 are introductoryand do not contain any new results. I have briefly described results, related withgroups, actions and simplicial homology that will be relevant for the sequel parts.In these chapters I have also introduced unified notations.

Chapter 4 is mostly based on papers [17, 21, 23]. In this chapter, I have explainedwhat modular homology is and how it can be used to estimate numbers of orbits.It also contains some of my own results. By studying modular homology, I havebeen able to improve one of results of the paper [17]. Section 4.7 contains fullproof of this improved result. Proposition 4.8.4 is also my own result.

The new results of the Thesis are concentrated in Chapters 5 and 6. In Chapter 5,I have used information about classical permutation groups from the Atlas [35]to explore situations when modular homology provides nontrivial results aboutnumbers of orbits. A typical example of the results proved in Section 5.2, is thefollowing:

Let G be any permutation group on a set Ω of cardinality 24. Sup-pose that the order of G is not divisible by 13, 17 and 19. Then Ghas at least 2 orbits on 6-subsets of Ω , at least 3 orbits on 8-subsetsof Ω and at least 4 orbits on 12-subsets of Ω .

A number of similar results follow from tables in Section 5.2. Section 5.3 containssome general results that belong to me.

In Chapter 6, I have explored orbits under action of linear groups on k -dimensionalsubspaces. The chapter is divided into two sections. In the first section I lookedat some of the subgroups of general linear groups and in the second section, Iconsidered linear representations of classical groups from the Atlas [35].

A typical example of a result of Section 6.1 is:

Every subgroup of the general linear group GL8(2) of order not di-visible by 85 has at least 3 orbits on 4 -dimensional subspaces of the8-dimensional vector space V8(2) over GF(2) .

Similarly, the next result is typical for Section 6.2:

Any subgroup of GL10(2) , isomorphic to the alternating group A12 ,has at least 3 orbits on 3-subspaces and at least 4 orbits on 4-subspaces.

Considerable number of similar results follow immediately from tables in Chapter6.

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Chapter 2

Groups and Group Actions

2.1 Groups

Groups provide the basic algebraic model for many types of mathematical ob-jects, from groups of real and complex numbers, to groups of permutations, ma-trices and linear transformation. The concept of group first evolved as a result tosolve algebraic equations by means of radicals. Today groups are not exclusivelyassociated with algebraic equations but it also appears in many different areasof mathematics.

Definition 2.1.1 Let X be a non-empty set. A binary operation on X ,denoted by ∗ , is a function from the cartesian product X ×X into X . That isa rule that associates to each ordered pair (a, b) of elements in X, a well definedelement a ∗ b in X .

Some familiar examples of binary operations on Z , the set of integers, are addi-tion, subtraction and multiplication. Division, however is not a binary operationon the set Z . This is because for some integers, a and b , a|b is not in Z .

Definition 2.1.2 A group is a set G , together with a binary operation ∗that satisfies the following conditions:

1. The binary operation ∗ is associative. That is, (a ∗ b) ∗ c = a ∗ (b ∗ c) forevery a, b, c ∈ G.

2. There is an identity element e ∈ G , such that g ∗ e = e ∗ g = g , for everyg ∈ G .

3. For every g ∈ G , there exists an inverse element, g′ , such that g ∗ g′ =g′ ∗ g = e

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In a group, if the binary operation is commutative, then the group is called anabelian group. The number of elements in a group is the order of the group andis denoted by the symbol |G| . A finite group is a group with a finite number ofelements.

Example 2.1.3 Let R� = R \ {0} be the set of all real numbers except zero.

Then R� , together with multiplication, forms a group G = 〈R�, •〉 . It is easy

to check that all the three group axioms are satisfied.

• The multiplication of real numbers is associative

• There is an identity element 1 ∈ R�

• For every element a ∈ R� , there is an inverse element a′ = 1

a∈ R

� suchthat a× a′ = 1

In fact, G = 〈R�, •〉 is an abelian group because multiplication in real numbersis commutative. Note however that the system 〈R, •〉 is not a group since 0 ∈ R

has no inverse in R

An important group could be constructed by having matrices as elements

Example 2.1.4 Let Mn(R) be the set of all square (n × n) matrices withentries in the field of real numbers R . The set of all invertible matrices fromMn(R) , with the binary operation of matrix multiplication forms a group. Thematrix multiplication is associative. The identity element of this group is the(n × n) identity matrix. All the matrices will have inverses since they are in-vertible. This group is however is not abelian since matrix multiplication is notcommutative.

Definition 2.1.5 The group of all invertible (n × n) matrices described inthe previous example is called the general linear group of degree n and is denotedby GLn(R) .

It is known that there exits a unique field (up to isomorphism) of order q = pk ,where p is a prime and k is some positive integer. This field is denoted byGF(q) . The set of all invertible n × n matrices over GF(q) , is denoted byGLn(q) . Furthermore the order GLn(q) ,

|GLn(q)| = (qn − 1)(qn − q)(qn − q2)qn − q3) · · · (qn − qn−1).

There are groups which are structurally alike to other groups. For example, lookat the two group tables below

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∗ 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2

• a b c da a b c db b c d ac c d a bd d a b c

The elements and the binary operation in each system is different. However, ifwe replace a with 0, b with 1, c with 2, d with 3 and • with ∗ , the secondtable will look exactly like the first table. So here we can make one table lookexactly like the other by renaming the elements and the binary operation. Thusthe structure of both groups is the same.

Definition 2.1.6 Let G and H be groups. A group isomorphism from G toH is a function f : G→ H that satisfies the following two conditions.

1. f is a one-to-one and onto function

2. f(g1g2) = f(g1)f(g2) for every g1, g2 ∈ G

2.1.1 Finite Groups

Finite groups have a finite number of elements. A group with only one element,namely the identity, satisfies all group axioms. This group is called the trivialgroup or the identity group.

The structure of a group of two elements, G = {e, a} is given by the table belowand there is only one such structure up to isomorphism.

∗ e ae e aa a e

There is also only one structure up to isomorphism of a group of three elements,G = {e, a, b} . This structure is given by the table below

∗ e a be e a ba a b eb b e a

A group of four elements has two possible group structures up to isomorphism.These are:

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∗ e a b ce e a b ca a b c eb b c e ac c e a b

and

∗ e a b ce e a b ca a e c bb b c e ac c b a e

The first table corresponds to the cyclic group, Z4 , while the second one definesthe Klein 4-group V .

The table below shows the number of group structures up to isomorphism ofdifferent orders up to 15.

Order Groups Abelian Non-Abelian1 1 1 02 1 1 03 1 1 04 2 2 05 1 1 06 2 1 17 1 1 08 5 3 29 2 2 010 2 1 111 1 1 012 5 2 313 1 1 014 2 1 115 1 1 0

From the table above, it is easy to note that abelian groups of all orders existwhile non-abelian groups do not exist for some orders.

2.1.2 Subgroups

Definition 2.1.7 Let G be a group and H be a non-empty subset of G .Then H is a subgroup if the following conditions are satisfied.

1. H is closed under the binary operation of G . That is, for every h1, h2 ∈H , also h1h2 ∈ H

2. H is closed under inverses, that is if h ∈ H , then h′ ∈ H

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Every group G contains at least two subgroups:

• The trivial subgroup {e}• G itself (the improper subgroup).

The other subgroups are called proper and non-trivial subgroups.

Example 2.1.8 Consider the Klein-4 group G = 〈{e, a, b, c}, ∗〉 , defined bythe previous table. The subsets H1 = {e, a}, H2 = {e, b} H3 = {e, c} allform subgroups of G , isomorphic to Z2

Definition 2.1.9 Let F be any field (possibly finite). Subgroups of the gen-eral linear group GLn(F) are called linear groups of degree n over the fieldF .

So linear groups are just groups of matrices. I will study some properties of lineargroups in Chapter 5.

2.1.3 Cyclic Groups

An important class of groups is called the cyclic group. The next fact could beeasily proved.

Proposition 2.1.10 Let G be a group. For g ∈ G let 〈g〉 = {gn ∈ G : n ∈ Z} .Then 〈g〉 is a subgroup of G.

Definition 2.1.11 The subgroup 〈g〉 is called the cyclic subgroup of G gen-erated by g . If 〈g〉 = G for some g ∈ G , we say that G is the cyclic groupgenerated by g. The element g ∈ G which generates G is called the generator ofG.

All elements present in a cyclic group are powers of the generator. Cyclic groupsof every possible powers exist.

Proposition 2.1.12 Let G be any cyclic group. Then

1. G is abelian

2. Every subgroup of G is cyclic.

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A set of n positive integers together with the binary operation of addition modulon forms the cyclic group of order n , denoted by Zn . For example if n = 4 ,then the table below gives the structure of Z4 .

⊕ 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2

The set of n roots of the equation zn = 1 also forms the cyclic group Zn undermultiplication. For example, let n = 4 , then the four roots of the equationz4 = 1 , are 1,-1, i and −i . These four roots also form the cyclic group Z4 .

Geometrically we may view the elements of Zn as n points spread out evenlyon the unit circle. So one can say that Zn is the group of rotations of a regularn-gon.

2.1.4 Permutation Groups and Orbits

Permutation groups are important class of groups whose elements are functionscalled permutations. As the next example shows, permutation multiplication isnot commutative.

Definition 2.1.13 Let X be a non-empty set. A permutation of X is aone-to-one and onto function φ : X → X. Composition of functions definesoperation of permutations called multiplication

The multiplication of permutations can be best explained by an example

Example 2.1.14 Let X = {1, 2, 3, 4, 5}. Then

φ1 =

(1 2 3 4 53 2 1 5 4

)and φ2 =

(1 2 3 4 54 5 3 1 2

)are two permutations of X .

Note that permutations can be multiplied as follows:

φ1φ2 =

(1 2 3 4 53 2 1 5 4

)(1 2 3 4 54 5 3 1 2

)=

(1 2 3 4 55 4 1 3 2

)

φ2φ1 =

(1 2 3 4 54 5 3 1 2

)(1 2 3 4 53 2 1 5 4

)=

(1 2 3 4 53 5 4 2 1

)So multiplication of permutation is not commutative.

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Proposition 2.1.15 Let X be a set. Then the set of all permutations of theset X forms a group under permutation multiplication.

Definition 2.1.16 The set of all n! permutations of a set X together withthe operation of permutation multiplication, is called the symmetric group andis denoted by Sn . Subgroups of Sn are called permutation groups.

Groups could be used to measure symmetry. With every geometric figure agroup is associated which characterizes the symmetry of the figure. For example,symmetry of a regular n-gon is characterized by the dihedral group denoted byDn . The order of Dn is 2n . Note that Dn is a subgroup of Sn

Example 2.1.17 The group of symmetries of a square is D4 . It has eightpermutations as its elements. These permutations describe the four rotationsand four reflections.

Example 2.1.18 S4 is the group of all symmetries of a 3-dimensional tetrahe-dra. In general Sn+1 is the group of symmetries of an n -dimensional tetrahedra.

Each permutation φ of a set determines a natural partition of X into cells withthe property that a, b ∈ X are in the same cell if and only if b = φn(a) for somen ∈ Z . This partition could be stated using an equivalence relation as follows:

For a, b ∈ X , let a ∼ b if an if only if b = φn(a) for some n ∈ Z

Definition 2.1.19 Let φ be a permutation of a set X . The equivalenceclasses in X determined by the equivalence relation above are the orbits of φ

Definition 2.1.20 A permutation φ ∈ Sn is a cycle if it has at most oneorbit containing more than one element. The length of a cycle is the number ofelements in the largest orbit.

Example 2.1.21 Let φ =

(1 2 3 4 5 63 2 6 5 4 1

)be a permutation. The

permutation φ is a cycle since there is one orbit with three elements and threefixed elements and we could be write φ = (1, 3, 6) = (3, 6, 1) = (6, 1, 3) . Howeverφ = (1, 6, 3) . ( The elements 2, 4 and 5 are fixed elements they do not appearwhen φ is written as a cycle.)

Proposition 2.1.22 Every permutation of a finite set is a product of disjointcycles.

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Example 2.1.23 φ =

(1 2 3 4 5 6 7 83 8 6 7 4 1 5 2

)= (136)(475)(28)

Definition 2.1.24 A cycle of length 2 is called a transposition.

It is known that every cycle is a product of transpositions and since every per-mutations can be written as cycles, then every permutation can be written as aproduct of transpositions.

Example 2.1.25

(1 2 3 4 5 6 7 83 8 6 7 4 1 5 2

)= (136)(475)(28) = (16)(13)(45)(47)(28)

Writing a cycle or permutation as a product of transpositions is not unique. Forexample

(123) = (13)(12)= (23)(12)(13)(23)= (23)(12)(13)(23)(12)(12)

The cycle (123) may also be written as a product of other even number (such as8, 10 or 12) of transpositions.

Proposition 2.1.26 No permutation in Sn can be expressed both as a productof an even number of transpositions and an odd number of transpositions.

So the number of transpositions used to represent a given permutation mustalways be even or always be odd.

Definition 2.1.27 A permutation of a finite set is called even if it may bewritten as a product of an even number of transpositions. Otherwise it is calledodd.

Definition 2.1.28 Let An be the set of all even permutations from Sn . ThenAn forms a group called the alternating group.

The order of An is n!2

and An may be viewed as the group of all rotations ofan (n− 1) -dimensional tetrahedra.

2.1.5 Normal Subgroups and Simple Groups

Definition 2.1.29 Let g be an element in a group G and let gH = {gh ∈G : h ∈ H} . The set gH is called the left coset of H in G determined by theelement g . The collection of all left cosets of H in G is denoted by G/H . Thenumber of left cosets of H in G is called the index of H in G and is denotedby the symbol [G : H] . Thus , [G : H] = |G/H|

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Definition 2.1.30 A subgroup H of a group G is normal if its left and rightcosets coincide. This means gH = Hg for all g ∈ G . We indicate that H is anormal subgroup of G by writing H � G .

Note that all subgroups of abelian groups are normal. Thus, all subgroups of theKlein 4-group are normal. An is also a normal subgroup of Sn .

Definition 2.1.31 A group is simple if it does not contain proper, non-trivialnormal subgroup.

Simple groups function as the building blocks of other groups. Finding all thesimple groups was a difficult task and it was completed in 1980 by a joint effortof hundreds of mathematicians.

Some of the simple groups are :

1. The cyclic group Zp , where p is a prime number.

2. The alternating group An for n ≥ 5 .

3. The 26 simple sporadic groups

As stated previously, sporadic groups are a family of simple groups. There are26 of these and they originated from the study of combinatorial objects. Thetable below lists the sporadic groups and their orders

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Name Denoted by OrderMathieu M11 24 · 32 · 5 · 11Mathieu M12 26 · 33 · 5 · 11Mathieu M22 27 · 32 · 5 · 7 · 11Mathieu M23 27 · 32 · 5 · 7 · 11 · 23Mathieu M24 210 · 33 · 5 · 7 · 11 · 23Janko J1 23 · 3 · 5 · 7 · 11 · 19Hall-Janko J2 27 · 33 · 52 · 7Janko J3 27 · 35 · 5 · 17 · 19Janko J4 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43Conway Co1 221 · 39 · 54 · 72 · 11 · 13 · 23Conway Co2 218 · 36 · 53 · 7 · 11 · 23Conway Co3 210 · 37 · 53 · 7 · 11 · 23Fischer Fi22 217 · 39 · 52 · 7 · 11 · 13Fischer Fi23 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23Fischer Fi24 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29Fischer-Griess F1 246 · 320 · 59 · 76 · 112 · 133 · 17 · ·19 · 23 · 29 · 31 · 41 · 47 · 59 · 59 · 71Fischer F2 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47Thompson F3 215 · 310 · 53 · 72 · 13 · 19 · 31Harada-Norton F5 214 · 36 · 56 · 7 · 11 · 19Higman-Sims HS 29 · 32 · 53 · 7 · 11McLaughlin Mc 27 · 36 · 53 · 7 · 11Suzuki Suz 213 · 37 · 52 · 7 · 11 · 13Lyons Ly 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67Held He 210 · 33 · 52 · 73 · 17Rudvalis Ru 214 · 33 · 53 · 7 · 13 · 29O’Nan O’N 29 · 34 · 5 · 73 · 11 · 19 · 31

2.1.6 Group Homomorphisms

A group homomorphism is a structure relating map between two groups. Suchmaps give us information about structure of a group from the known structuralproperties of the other groups.

Definition 2.1.32 Let G and H be groups. A group homomorphism is afunction ψ : G→ H such that ψ(g1g2) = ψ(g1)ψ(g2) for all g1, g2 ∈ G .

Example 2.1.33 Let Sn be the group of all permutations of n letters, and letσ and μ be any permutation from Sn . We could define a function ψ : Sn → Z2

as

ψ(σ) = 0 , if σ is an even permutation.

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ψ(σ) = 1 , if σ is an odd permutation.

It is easy to check that ψ(σμ) = ψ(σ) + ψ(μ)(mod2) and thus ψ defines ahomomorphism of Sn into Z2 .

For any group G and H , there is at least one homomorphism, ψ : G → Hnamely the trivial homomorphism, defined by ψ(g) = e′ for all g ∈ G . (Heree’ is the identity element of H .) Now let us look at some of the properties ofgroup homomorphisms. To do this we need a set theoretic definitions. Note thatsquare brackets are used when a function is applied to a subset of its domain.

Definition 2.1.34 Let ψ be a mapping of a set X into a set Y , let A ⊆ Xand B ⊆ Y . Then the image ψ[A] of A in Y under ψ is {ψ(a) : a ∈ A} .The set ψ[A] is sometimes called the range of ψ . The inverse imageψ−1[B] ofB in X is {x ∈ X : ψ(x) ∈ B}

Proposition 2.1.35 Let ψ be a homomorphism of a group G into group H.Then the following properties hold.

1. If e is the identity in G, then ψ(e) is the identity e′ in H.

2. If g ∈ G , then ψ(a−1) = ψ(a)−1 .

3. If K is a subgroup of G, then ψ[K] is a subgroup of H.

4. If L is a subgroup of H, then ψ−1[L] is a subgroup of G.

Informally speaking, ψ preserves the identity, inverses and subgroups.

For a group homomorphism ψ : G → H , let e be the identity element of Gand e′ be the identity element of H . Since {e′} is a subgroup of H , it followsfrom the preceding proposition that ψ−1[e′] is a subgroup of G .

Definition 2.1.36 Let ψ : G → H be a homomorphism of groups. The sub-group ψ−1[e′] = {g ∈ G : ψ(g) = e′} is the kernel of ψ denoted by Ker (ψ)

Example 2.1.37 Consider the homomorphism of Sn into Z2 in example (2.1.33). All the even permutations form a subgroup in Sn while the image of all theeven permutations, is the identity ,which is a subgroup in Z2 . The kernel of thishomomorphism is the alternating group An .

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2.2 Group actions

Let Ω be a set of objects and G is a group. Informally speaking group action ofG on Ω may be considered as ’multiplying’ elements α of Ω by elements g ofG such that product gα is again in Ω . Under group action an element α1 ofΩ may be sent to another element α2 of Ω . The notation gα1 = α2 ( g ∈ G ,α1, α2 ∈ Ω ) means that an element g acts on α1 and sends it to α2

Definition 2.2.1 Let Ω be a set and G a group. An action of G on Ω is amap · : G× Ω → Ω such that

1. ex = x for all x ∈ Ω. (here e is the identity of G )

2. (g1g2)x = g1(g2x) for all x ∈ Ω and g1, g2 ∈ G

Under these conditions, Ω is called a G-set.

Suppose that gx = gy for some g ∈ G and x, y ∈ Ω, then from the sec-ond condition in the definition above, we have g−1(gx) = g−1(gy). From this(g−1g)x = (g−1g)y or x = y. This means that each element of G permutes theset Ω . Thus one may associate a permutation of Ω to every element of group Gand group action could be viewed as a homomorphism of G into SΩ , where SΩ

is the set of all permutations of Ω . This homomorphism is called the permuta-tion representation of G. If the identity element of G is the only element whichleaves all the elements of Ω fixed, then the group action is said to be faithful.

For every α ∈ Ω an important subgroup Gα of G called the stabilizer of α ∈ Ωis defined as

Gα := {g : g(α) = α}

A subset {α : g(α) = α} of all elements of Ω fixed by g is denoted by Ωg.Both Gα and Ωg will be used in the next section.

2.2.1 Orbits

For a group G acting on a set Ω , the orbit of α ∈ Ω is a subset of Ω whichcontains all the elements to which α can be moved to under the action of G .The orbit containing α is denoted orbG(α) . Thus,

orbG(α) = {gα : g ∈ G}We will show now that the orbits of Ω are disjoint, thus an element of Ω canbelong to precisely one orbit. So Ω is a union of all its orbits.

18

Proposition 2.2.2 Let Ω = {α0, α1, . . . , αn} be a set and let G be a group.The orbits are equivalence classes of the relation ∼ . Here αi ∼ αj if there existsa g ∈ G such that gαi = αj , for αi, αj ∈ Ω

Proof. For every α ∈ Ω and e ∈ G , we have eα = α , satisfying reflexiveproperty. Also if α1 ∼ α2 , we have gαi = αj then αi = g−1αj . Thus αj ∼ αi

satisfying the symmetric property. If αi ∼ αj and αj ∼ αk , then gαi = αj andhαj = αk for some g, h ∈ G . We may write αj ∼ αk as hgαi = αk , satisfyingthe transitive property. �

Definition 2.2.3 An action of a group G on a set Ω is said to be transitiveif there is only one orbit or equivalently, for every αi, αj ∈ Ω , there exists g ∈ Gsuch that gαi = αj .

Definition 2.2.4 Let Ω be a set of cardinality n . The set of all k -tuplesof Ω is denoted by Ωk (for k ≤ n ). The action of G on Ω is said to bek -transitive if there is only one orbit of G on Ωk . The set of all k -subsets ofthe set Ω is denoted Ω{k} . The action of G on Ω is said to be k -homogeneousif there is only one orbit of G on Ω{k} . The orbits of G on Ω{k} are calledsymmetrized k -orbits. I will call symmetrized k -orbits briefly just by k -orbits.

Evidently, group which are k -transitive are also k -homogeneous.

Since this thesis is centered around group actions and orbits, I will give somedetailed examples.

Example 2.2.5 Let G be a group and for every g ∈ G let φg : G → G be amap given by φg(x) = gxg−1 , for x ∈ G . Now I can try to define an action ofG on the set X of all elements of G by the rule g(x) = φg(x) = gxg−1 . Toprove that it is indeed an action, I need to show that (gh)(x) = g(h(x)) for allg, h, x ∈ G . Indeed,

(gh)(x) = φgh(x) = (gh)x(gh)−1 = (gh)x(h−1g−1) = g(hxh−1)g−1 =gφh(x)g

−1 = φg(φh(x)) = g(h(x)).

Also if e is the identity element in G then for every x ∈ G , e(x) = e−1xe = x .Thus, the map φ is an action of G on itself. The orbits under this action arecalled conjugacy classes of the group G .

Example 2.2.6 We will look at actions of four different groups on the set Ω ={α1, α2, α3, α4}. The four groups are:

G1 =

{(1 2 3 41 2 3 4

)}� {e}

19

G2 =

{(1 2 3 41 2 3 4

) (1 2 3 44 2 3 1

)}� Z2

G3 =

{(1 2 3 41 2 3 4

) (1 2 3 44 3 2 1

)}� Z2

G4 =

{(1 2 3 41 2 3 4

) (1 2 3 42 3 4 1

) (1 2 3 43 4 1 2

) (1 2 3 44 1 2 3

)}� Z4

The action is defined by

gαi = αg(i), (g ∈ G)

All the elements of Ω are moved to themselves by G1 , hence there are fourorbits, with each orbit containing only one element.

The group G2 moves α1 into α4 , α2 into α2 , α3 into α3 and α4 into α1 . Thusthe group G2 has three orbits. These are orbG(α1) = {α1, α4} orbG(α2) = {α2}and orbG(α3) = {α3} .

The group G3 has two orbits, orbG(α1) = {α1, α4} and orbG(α2) = {α2, α3} .Lastly the group G4 has only one orbit on Ω . This orbit could be denoted aseither orbG(α1) , orbG(α2) , orbG(α3) or orbG(α4) .

Note that if we consider the set Ω to be the set of vertices of a square, thenG1 is the group that fixes all the four points. G2 flips the square about thediagonal through points α2 and α3 . The group G3 is that flips the squareabout its vertical axis of symmetry and G4 rotates the square through an angleof 90 degrees.

Example 2.2.7 Consider the action of the dihedral group D4 on the square

� �

� �

�

��

��

��

��

��

��

���

��

��

��

��

��

��

���

� �

�

�A C

EG

H

F

B

D

Some observation will show that none of the eight elements of D4 would movepoint B into C . Likewise D will never be moved to E . In fact there are twoorbits, orbG(A) = {A,C,E,G} and orbG(B) = {B,D, F,H}

Later in this thesis, we would study actions of permutation groups and lineargroups on subsets and on subspaces respectively.

20

2.2.2 Enumeration of Orbits

In this section we would look at two famous results which count the number oforbits. These results are known as Burnsides Lemma and Polya’s Theorem

The Burnside’s lemma stated below provides a method of finding the number oforbits, N(G) := N , on a set Ω under an action of a group G .

Proposition 2.2.8 (The Burnside’s Lemma) Let G be a permutationgroup of degree n acts on a set Ω of cardinality n . Let N(G) := N be thenumber of orbits of G on Ω , and Ωg be the subset of Ω fixed by the groupelement g . Then

N =1

G

∑g∈G

|Ωg|

Proof. Let Gx be the stabilizer of x and let x, y ∈ Ω belong to the sameorbit. Suppose that hx = y for some h ∈ G . Since gx = x for g ∈ Gx , wehave ghx = hx = y .

The set of all elements h such that hx = y is the coset hGx . The number ofcosets of Gx in G is equal to |orbG(x)| . The number of elements in a orbitmultiplied by the number of cosets equals the order of G . Thus

|orbG(x)||Gx| = |G| (2.1)

(Of course all cosets have the same number of elements and if x and y are inthe same orbit then |Gx| = |Gy| .)Now let us count all pairs (g, x) where g(x) = x , with g ∈ G and x ∈ Ω . LetK be the number of such pairs. For each g ∈ G , there are |Ωg| pairs having gas the first element. Thus

K =∑g∈G

|Ωg| (2.2)

On the other hand for each x ∈ Ω , there are |Gx| pairs having x as the secondmember. So we have

K =∑x∈Ω

|Gx| (2.3)

The right side of (2.3) is also equal to

|Gx1||orbG(x1)| + |Gx2||orbG(x2)| + · · · + |GxN||orbG(xN)|

Here xi, (i = 1, · · · , N) , are representatives from distinct orbits. To completethe proof we use the equation (2.1) and write

|Gx1||orbG(x1)|+ |Gx2||orbG(x2)|+ · · ·+ |GxN||orbG(xN)| = |G|+ |G|+ |G|+ · · ·

21

= N |G|Thus we have

N |G| =∑g∈G

|Xg|

or

N =1

|G|∑g∈G

|Xg|

�To state the Theorem of Polya we need to introduce first the notion of the cycleindex.

Every permutation g of a set Ω can be written as a product of disjoint cycles.If g has c1 cycles of length 1, C2 cycles of length 2, · · · , Cn cycles of length n ,then the monomial xc1

1 xc22 . . . xcn

n in the indeterminates x1, x2, . . . , xn is calledthe cycle index of the permutation g .

If G is a permutation group on Ω , then the cycle index of G is the average ofthe cycle indices of its elements. That is, the cycle index of a group G is

Z(G;x1, x2, . . . , xn) =1

|G|∑g∈G

xc1(g)1 x

c2(g)2 . . . xcn(g)

n

Here ci(g) is the number of cycles of length i of g .

Example 2.2.9 The table below shows the elements of the permutation groupZ6 = 〈(1, 2, 3, 4, 5, 6)〉 and their cycle indices.

permutation cycle index(1 2 3 4 5 61 2 3 4 5 6

)x6

1(1 2 3 4 5 62 3 4 5 6 1

)x6(

1 2 3 4 5 63 4 5 6 1 2

)x2

3(1 2 3 4 5 64 5 6 1 2 3

)x3

2(1 2 3 4 5 65 6 1 2 3 4

)x2

3(1 2 3 4 5 66 1 3 3 4 5

)x6

The cycle index of the group Z6 is thus

Z(Z6;x1, x2, x3, x4, x5, x6) =1

6(x6

1 + x32 + 2x2

3 + 2x6)

The next important result can be used to find the number of orbits of G on thek -subsets of a set Ω .

22

Theorem 2.2.10 (The Polya’s Theorem) Let Ω be a set of cardinal-ity n . The number of orbits of G on the k -subsets of Ω , (k ≤ n) , is equal tothe coefficient of yk in

Z(G; 1 + y, 1 + y2, . . . , 1 + yn)

The above proposition is one of the many corollaries of the more general Polya’stheorem, that could be used to enumerate more complicated objects.

Example 2.2.11 Let Ω be a set of six elements and let the group Z6 act on allk -subsets of Ω , for 0 ≤ k ≤ 6 . The number of orbits of Z6 on the k -subsets isgiven by the coefficient of yk in

1

6

[(1 + y)6 + (1 + y2)3 + 2(1 + y3)2 + 2(1 + y6)

]expanding and simplifying this we get

1

6

[6 + 6y + 18y2 + 24y3 + 18y4 + 6y5 + 6y6

]= 1 + y + 3y2 + 4y3 + 3y4 + y5 + y6 (2.4)

By looking at the coefficients of yk in (2.4) we have the following results:

one orbit on 0-subset, one orbit on 1-subsets, three orbit on 2-subsets, four orbiton 3-subsets, three orbit on 4-subsets, one orbit on 5-subsets and one orbit on6-subsets.

In this thesis we consider estimating the number of orbits when a permutationgroup acts on the set of subsets and also when a linear group acts on a set ofsubspaces.

Another famous result about k -orbits is called the inequalities of Livingstoneand Wagner. It states that if a permutation group acts on the set of all thesubsets of a finite set, then the numbers of orbits form a unimodal sequence.

Theorem 2.2.12 (The Inequalities of Livingstone and Wagner) Letthe cardinality of a finite set Ω be n , and set m = n

2if n is even and m = n−1

2,

if n is odd. Also let Nk := Nk(G) denote the number of orbits of a permutationgroup on k -subsets of Ω . Then

Nm ≥ Nm−1 ≥ Nm−2 ≥ · · · ≥ N0

Also Nm−i = Nm+i if n is even, and Nm−i = Nm+1+i if n is odd.

The same inequalities hold in case of linear groups acting on the set of all sub-spaces of a finite vector space. In this case n is the dimension of the finite vectorspace.

23

2.3 Group Representations

A representation of a group G is a way of visualizing a group as a group ofmatrices. Thus with each element of G a matrix is associated. In this section,F is the field of complex numbers.

Definition 2.3.1 Let G be a group and F be a field. Let GLn(F) be agroup of invertible n× n matrices, described in section (2.1) . A representationof G over F is a homomorphism ρ from G to GLn(F) , for some n . Theinteger n is the degree of the representation.

Since representations are homomorphisms, for any representation ρ : G →GLn(F) , we have

• ρ(gh) = (ρ(g))(ρ(h)) (for g, h ∈ G )

• ρ(e) = In (Here e is the identity element of G and In is the identity

matrix in GLn(F) )

• ρ(g−1) = (ρ(g))−1

Definition 2.3.2 The representation ρ : G → GL1(F) which is defined byρ(g) = (1) for all g ∈ G is called the trivial representation.

In other words, the trivial representation of a group G is the representationwhere every element of G is mapped to the 1 × 1 identity matrix.

Important class of representations are faithful representations. In this represen-tation, each matrix corresponds to exactly one element of the group. Thus, afaithful representation is injective.

For groups G and H , to show that a map φ : G → H is an injection we haveto show that Ker (φ) = {e}, where e is the identity of G . Thus we have thefollowing

Definition 2.3.3 A representation ρ : G→ GLn(F) is faithful if and only ifKer (ρ) = {e}

Definition 2.3.4 Let ρ : G → GLm(F) and σ : G → GLn(F) be rep-resentations of G over F . Then ρ is equivalent to σ if n = m and thereexists an invertible n × n matrix T ∈ GLn(F) such that for all g ∈ G holdsσ(g) = T−1(ρ(g))T .

24

There could be a lot of representations equivalent to any given representation.

Definition 2.3.5 A representation of degree n of a group G is reducible (overF ) if it is equivalent to a representation of the form

g →(Xg 0Yg Zg

), g ∈ G (2.5)

for some matrices Xg , Yg and Zg , where Xg is of size k × k and 0 < k < n.

For g ∈ G , the functions g → Xg and g → Zg are also representations. Thusnow there are two representations of smaller degree.

If a representation of degree n is not equivalent to any representation in theform (2.5), then the representation is irreducible. This means that the smallestmatrices that could be used to represent the elements of the group would ben× n . The representations of degree one are of course irreducible.

While there are many representation of a group, the number of irreducible rep-resentations are fixed and it is equal to the number of conjugacy classes of agroup.

When representations of a group are G are over the field of complex numbersC (or over any finite field of characteristic co-prime to the order of G ), thefamous Maschke’s Theorem states that the irreducible representations are like thebuilding blocks of all other representations. In other words every representationρ of a group G can be expressed in the form

g →

⎛⎜⎜⎜⎜⎜⎜⎝

Ag 0 0 0 0 00 Bg 0 0 0 00 0 · 0 0 00 0 0 · 0 00 0 0 0 · 00 0 0 0 0 Ng

⎞⎟⎟⎟⎟⎟⎟⎠

(g ∈ G)

where Ag , Bg , . . . , Ng are irreducible representations of G .

Let Ug be any irreducible representation of G , and let c(U) be the number ofthose representations Ag , Bg , . . . , Ng that are equivalent to Ug . The numberc(U) is called the multiplicity of the representation U . I shall use multiplicitiesof representations in Chapter 3.

25

2.4 Finite Linear Spaces

Just as groups of permutations naturally act on subsets, linear groups naturallyact on subspaces of a vector space. In this Section I will describe some propertiesof such subspaces I will need later.

The n -tuples (for some positive n ) with entries from GF(q) , form vectors ofthe n -dimensional vector space Vn(q) . For each component in the n -tuples,there are q choices. Thus there are qn vectors in Vn(q) . The number ofk -dimensional subspaces of Vn(q) is given by the Gaussian coefficient

[nk

]q

:=(qn − 1)(qn−1 − 1) . . . (qn−k+1 − 1)

(qk − 1)(qk−1 − 1) . . . (q − 1)

Note that the number of k -subsets of a set of cardinality n is given by thebinomial coefficient (

nk

):=

n!

k!(n− k)!

Example 2.4.1 Let n = 3 and q = 2 . Then V3(2) is the 3-dimensionalvector space over GF(2) . There are 23 vectors in the space V3(2) and thesevectors are (0, 0, 0) , (0, 0, 1) , (0, 1, 0) , (0, 1, 1) , (1, 0, 0) , (1, 1, 0) , (1, 0, 1) and(1, 1, 1) .

The number of 2-dimensional subspaces of V3(2) is given by the Gaussian coef-ficient [

32

]2

= 7

and they are:

〈(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)〉〈(0, 0, 0), (1, 0, 0), (0, 0, 1), (1, 0, 1)〉〈(0, 0, 0), (0, 1, 0), (0, 1, 1), (0, 1, 1)〉〈(0, 0, 0), (1, 0, 0), (1, 1, 1), (0, 1, 1)〉〈(0, 0, 0), (0, 1, 0), (1, 1, 1), (1, 0, 1)〉〈(0, 0, 0), (0, 0, 1), (1, 1, 1), (1, 1, 0)〉〈(0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)〉

Note that the notation 〈a1, a2, · · · , an〉 denotes the vector space spanned by thevectors a1, a2, · · · , an .

26

Similarly, it could be easily verified that the number of 3-dimensional subspacesof V4(3) is equal to [

43

]3

=(34 − 1)(33 − 1)(32 − 1)

(33 − 1)(32 − 1)(3 − 1)= 40

27

Chapter 3

Homology Theory

A homology theory is a method that associates with a topological space X analgebraic object that expresses some of the topological properties of X . Much ofthe homology theory was developed by Henry Poincare in 1890’s in an attemptto associate with surfaces some algebraic objects that would be invariant undertopological transformations. Two surfaces are considered homologically equiv-alent if both surfaces have the same set of these algebraic objects attached tothem.

Homology theory is based on properties of some sequences of vector spaces (or,more generally, modules) called homological sequences. In this thesis I will useonly homological sequences of vector spaces. I begin by looking at such sequences.

3.1 The Homological Sequence

Let F be any field. Let V0 , V1 , V2 ,. . . , Vn be vector spaces over the field F

and let ∂1 , ∂2 , . . . , ∂n be linear maps, such that ∂i : Vi → Vi−1 , (i = 1, . . . , n) .Thus, we have the following sequence

V0∂1←− V1

∂2←− V2∂3←− · · · ∂n−1←− Vn−1

∂n←− Vn (∗)

Definition 3.1.1 The sequence above is called homological if

Im ∂m ⊆ Ker ∂m−1

for every m = 1, 2, . . . , n . The sequence is called exact if

Ker ∂m−1 = Im ∂m

The factor spaceHm = Ker ∂m−1/Im ∂m

28

is called the m-th homology module or simply the m-th homology of the sequence(∗) over the field F .

Thus, any exact sequence is homological, but not every homological sequence isexact. It could be also said that the sequence (∗) is homological if compositionof every two consecutive arrows is zero, i.e. ∂m−1∂m = 0 , (m = 0, 1, . . . , n) .

Since I am looking at homology over fields, the homology modules Hm occursto be vector spaces and so we may talk about their dimensions. The dimensionof Hm is called the m -th Betti number and is denoted as

βm = dimHm = dim Ker ∂m−1 − dim Im ∂m.

For a homological sequence

· · · ←Mk−1 ←Mk ←Mk+1 ←Mk+2 ← · · · (12)

the number

χ =+∞∑

k=−∞(−1)k dimMk

is called the Euler characteristic of the sequence.

Another important equation associated with a homological sequence is the Hopf-Lefschetz formula stated below. Before looking at this formula let us recall whatthe trace of a matrix is. Let A = (aij) be a n × n square matrix. Then thetrace of A , denoted as trA is defined by the equality

trA =n∑

i=1

aii

Proposition 3.1.2 (Hopf-Lefschetz formula)

Let G be a finite group and let

· · · ←Mk−1 ←Mk ←Mk+1 ←Mk+2 ← · · · (12)

be a homological sequence. Suppose that the vector spaces and the homologies inthis homological sequence are FG -modules. Then for the representations of Gon vector spaces Mk , and on the homologies Hk , (k ≥ 0) , holds:∑

k≥0

(−1)ktrMk =∑k≥0

(−1)ktrHk

29

If in the above proposition the group G is trivial then the Hopf-Lefschetz formulasimplifies to the Euler-Poincare equation:∑

k≥0

(−1)k dimMk =∑k≥0

(−1)k dimHk

This is because for each Mk and Hk the representation is a single identity matrixwhose size depends on the dimension of Mk and Hk respectively and taking thetrace gives us this dimensions.

We shall now look at an example of homology theory: the homology theory ofsimplicial complexes.

3.2 The Simplex and the Simplicial Complex

I will describe how homological sequences of vector spaces can be associatedwith surfaces (or other topological spaces). For this the topological space mustbe triangulated by a simplicial complex. I am explaining these notions below.

Let Ω = {α0, α1, α2, . . . , αn} be a finite set. Elements αi of Ω will be calledvertices. The set of all subsets of Ω is denoted by 2Ω . A subset Δ of 2Ω iscalled a simplicial complex on the vertex set Ω if whenever σ ∈ Δ belongs toΔ and τ ⊆ σ , then also τ ∈ Δ . In particular, the empty set ∅ always belongsto Δ .

The elements of Δ are called faces or simplices. In the case when Δ = 2Ω , Δis called the simplex on Ω and is denoted by Σn if |Ω| = n . The dimensionof σ ∈ Δ is defined by dim σ = |σ| − 1 . A (k − 1) -dimensional simplex or facewould be abbreviated as k -face or k -simplex. The dimension dim Δ of Δ isthe maximum of {dimσ : σ ∈ Δ} . A simplicial complex Δ is called pure if allmaximal faces have the same dimension. Maximal faces are also called facets ofthe complex.

The simplex on the set of vertices σ is similarly denoted as 2σ which is thecollection of all subsets of σ .

Example 3.2.1 The Figure below shows a 1-dimensional simplical complexΔ on the set Ω = {α0, α1, α2, α3} .

30

�

� �

�

��

��

��

��

��

��

��

��

��

��

��

��

α0 α3

α1

α2

Let P0 = {{α0}, {α1}, {α2, }{α3}} be the set of all 0 -dimensional faces of Δ .In other words, P0 is the set of all vertices or 1 -subsets of Ω .

Let P1 = {{α0α1}, {α0α2}, {α1α2}, {α1α3}, {α2α3}} be the set of 1 -dimensionalfaces of Δ . These 1 -dimensional faces of Δ are also called edges . P1 could alsobe viewed as a set of 2 -subsets of Ω . Similarly let P2 = {{α0α1α2}, {α1α2α3}}be the set of 2 -dimensional faces of Δ .

3.3 Homology of Simplicial Complexes

In this Section I will associate homology with simplicial complexes. For this Ineed to look at vector spaces spanned by faces of a complex. In this section Iwill assume that F is the field R of real numbers.

Definition 3.3.1 Let Δ be a simplicial complex of a vertex set Ω = {α0, α1 ,α2, . . . , αn−1} and |Ω| = n . The vector space over F with all k -faces of Δbeing a basis is denoted by Pk, ( k ≤ n ).

Let σ be a simplex. Define two orderings of its vertex set to be equivalent ifthey differ by an even permutation. If the dimension of σ its greater than zero,then the orderings of the vertices of σ fall into two equivalence classes. Each ofthis equivalence classes is called an orientation of σ . (If σ is a 0-simplex, thenthere is only one class and only one orientation of σ . An oriented simplex is asimplex σ together with an orientation of σ .

Example 3.3.2 The figure below shows an oriented 2-simplex.

31

� �

�

��

��

���

��

��

�α1 α2

α0

Note that α0α1α2 and α1α2α0 have the same orientation, however α0α1α2 doesnot have the same orientation as α0α2α1 . We write α0α1α2 = −α0α2α1 . Thenegative sign is used to indicate the opposite orientation.

The boundary of an orientated k -simplex is the sum of its orientated (k − 1) -faces. For example if we consider again the orientated 2-simplex in the previousexample, we may write an expression for the boundary of this simplex as

α0α1 + α1α2 + α2α0 = α1α2 − α0α2 + α0α1,

considering the orientation. Thus the boundary map is defined as follows

Definition 3.3.3 A linear map ∂k : Pk → Pk−1 is called the boundary map andis given by

∂k{α0, α1, α2, · · ·αk−1} =k−1∑i=0

(−1)i(α0, α1, α2, · · · αi · · ·αk−1)

Here τ = {α0, α1, α2, · · ·αk} is a k -simplex from the space Pk , while (α0 ,α1 ,α2 ,. . . , αi · · ·αk) is a (k − 1) -simplex from Pk−1 . The symbol αi meansthat we need to delete the vertex αi from the array. So the boundary operatormaps a k -simplex to alternating sum of (k − 1) -simplices.

Theorem 3.3.4 The sequence of boundary maps

P0∂1←− P1

∂2←− · · ·Pk∂k+1←− Pk+1 · · ·Pn−2

∂n−1←− Pn−1∂n←− Pn (k < n)

associated with the simplicial complex, on n vertices, is homological.

proof. Let σ be any k -face of the simplicial complex Δ . Then

∂k−1∂k(σ) = ∂k−1(k−1∑i=0

(−1)k(α0, α1, · · · , αi, · · ·αk−1))

= (k−1∑i=0

(−1)k∂k−1(α0, α1, · · · , αi, · · ·αk−1)

32

=k−1∑i=0

k−1∑i<j

(−1)i(−1)j−1(α0, α1, · · · , αi, · · · αj, · · ·αk−1)

+k−1∑i=0

k−1∑j<i

(−1)i(−1)j(α0, α1, · · · , αj, · · · αi, · · ·αk−1)

each face that occurs in the first sum also occurs in the second sum but withopposite sign. Therefore the two sums cancel each other. �

Example 3.3.5 Look at the simplicial complex Δ considered in the firstExample of this chapter. Let F be the field R . Then FP0 is a 4 -dimensionalvector space over F with the basis {α0}, {α1}, {α2}, {α3} .

Likewise, FP1 is a 5 -dimensional vector space spanned by {α0α1} , {α0α2} ,{α1α2} , {α1α3} , {α2α3} . Let FP2 be a 2 -dimensional vector space spannedby {α0α1α2} and {α1α2α3}} , so that an arbitrary element of FP2 is of the form

λ1α0α1α2 + λ2α1α2α3, (λ1, λ2 ∈ F).

Let ∂2 : FP2 → FP1 be the boundary map defined as

∂2(α0α1α2) = α0α1α2 − α0α1α2 + α0α1α2

= α1α2 − α0α2 + α0α1

and∂2(α1α2α3) = α1α2α3 − α1α2α3 + α1α2α3

= α2α3 − α1α3 + α1α2

So the matrix for ∂2 is

∂2 =

⎛⎜⎜⎜⎜⎝

1 0−1 0

1 10 −10 1

⎞⎟⎟⎟⎟⎠

Similarly ∂1 : FP1 → FP0 is defined by

∂1(α0α1) = α1 − α0 ∂1(α1α3) = α3 − α1

∂1(α0α2) = α2 − α0 ∂1(α2α3) = α3 − α2

∂1(α1α2) = α2 − α1

The matrix for ∂1 is

∂1 =

⎛⎜⎜⎝

−1 −1 0 0 01 0 −1 −1 00 1 1 0 −10 0 0 1 1

⎞⎟⎟⎠

33

As we can see,

∂1∂2 =

⎛⎜⎜⎝

−1 −1 0 0 01 0 −1 −1 00 1 1 0 −10 0 0 1 1

⎞⎟⎟⎠

⎛⎜⎜⎜⎜⎝

1 0−1 0

1 10 −10 1

⎞⎟⎟⎟⎟⎠ = 0

as it is required.

The Ker ∂1 = {x : ∂1x = 0, x ∈ FP1} is the space of solutions of the system

⎛⎜⎜⎝

−1 −1 0 0 01 0 −1 −1 00 1 1 0 −10 0 0 1 1

⎞⎟⎟⎠

⎛⎜⎜⎜⎜⎝

x1

x2

x3

x4

x5

⎞⎟⎟⎟⎟⎠ = 0,

Hence Ker ∂1 is a two dimensional space spanned by the vectors (1,−1, 1, 0, 0)T

and (1,−1, 0, 1, 1)T

The dim Im ∂2 is equal to the rank of the matrix for ∂2 . Since the two col-lum vectors of this matrix are linearly independent, the rank is 2, and thus thedim Im ∂2 = 2.

The Betti number β1 could now be calculated.

β1 = dim Ker ∂1 − dim Im ∂2

= 2 -2= 0

The dim Im ∂3 = 0 , since ∂3 is defined as ∂3 : 0 → FP2 . It can also be seenvery easily that Ker ∂2 = 0 . So

β1 = dimKer ∂2 − dimIm ∂3

= 0 − 0= 0

Thus, the homology of the complex Δ has been just calculated.

The homologies of simplicial complexes are homotopy invariants. This meansthat two topologically equivalent complexes have the same set of homologies.For example a solid cube is regarded topologically equivalent to the solid sphereand so these two objects will have the same homologies. I shall not describe thehomotopy map or homotopy invariants since this thesis mainly deals with thecombinatorial properties of the homological sequence rather than the topologicalproperties.

34

Chapter 4

Modular Homology Theory

In the previous chapter I considered usual homology associated with homologicalsequences where composition of any two consecutive arrows is equal to 0. Now Iwill look at sequences of vector spaces where composition of every N consecutivearrows is 0, (N > 2 ). It occurs that such sequences also could be used toproduce generalized homology. This kind of homology appears to be mentionedfirst in W Mayer [16] in 1942. Recently it has been used in papers C. Kassel andM. Wambst [13], M. Dubois-Violette [9], M. Kapranov [14], A. Tikaradze [33].A special case of generalized homology, called the modular homology has beenstudied by T. Bier [3] and V. Mnukhin and J. Siemons [17, 21].

4.1 Generalized Homology

I shall explain the idea of the generalized homology by the next Example first.

Consider the sequence

· · · ←Mn−3δ←Mn−2

δ←Mn−1δ←Mn ← 0

Suppose that this sequence has the property that δ3 = 0 . From this sequence,three subsequences could be constructed as follows:

· · · δ←Mn−5δ2←Mn−3

δ←Mn−2δ2←Mn ← 0

· · · δ2←Mn−4δ←Mn−3

δ2←Mn−1δ←Mn ← 0

· · · δ2←Mn−5δ←Mn−4

δ2←Mn−2δ←Mn−1 ← 0

Now each of these three subsequences has the property that composition of anytwo arrows is zero. Thus all the three of the subsequences are homological.

35

Definition 4.1.1 Let N be any integer greater that 2. A sequence

· · · δ←Mk−3δ←Mk−2

δ←Mk−1δ←Mk

δ←Mk+1δ←Mk+2

δ← · · · (4.1)

is called a generalized homological sequence if δN = 0 .

In the previous definition, if N = 2 , then we have the usual homological se-quence.

Subsequences of (4.1) of the form

· · · δN−i← Mk−iδi←Mk

δN−i← Mk+N−iδi← · · · (k, i ∈ Z, i < k)

are homological (that means that the composition of any two consecutive mapsis zero).

Note that a convention used here is that the generalized homological sequence isinfinite. This done by adding zero modules at the beginning and at the end ofthe sequence.

Definition 4.1.2 Let

· · · δ←Mk−3δ←Mk−2

δ←Mk−1δ←Mk

δ←Mk+1δ←Mk+2

δ← · · ·

be a generalized homological sequence with δ being an arbitrary linear map suchthat δN = 0 and let

· · · ←Mk−iδi←Mk

δp−i← Mk−i+p ← · · ·

be a homological subsequence.

The generalized homology at Mk in this homological sequence is denoted by

Hk,i = (Ker δi ∩Mk)/(Im δp−i ∩Mk) .

The dimension of Hk,i is the Betti number, denoted by βk,i

Definition 4.1.3 A generalized homological sequence is called exact if all itsBetti numbers are zero. In other words if all the corresponding homologies aretrivial. A generalized homological sequence is called almost exact if at most oneof the Betti numbers is non-zero in each of the homological subsequences.

36

4.2 Composition of Inclusion Maps

In the previous chapter, the standard simplical homology on a simplicial complexΔ has been considered. The simplical homology is concerned with the boundarymap ∂ , which maps a k -faces of a simplex to an alternating sum of (k−1) -faces.This is equivalent to saying that the operator ∂ maps k -subsets of a finite setΩ , ( |Ω| = n , k ≤ n ) to the alternating sum of (k − 1) -subsets of Ω .

Modular homology, which I shall describe in the next section, is concerned witha different linear map, δ , defined as follows.

Definition 4.2.1 Let τ be a k -dimensional face {α0, α1, α2, · · ·αk} of a sim-plex Σn . Then

δτ =k∑

i=0

{α0, α1, α2, · · · , αi, · · · , αk}

The symbol αi means that we need to delete the vertex αi from the array. Themap δ is called the inclusion map.

The notation δk is used to specifically denote the inclusion map from the spacespanned by k -dimensional faces to the space spanned by (k − 1) - dimensionalfaces. The symbol δ will be used to denote arbitrary inclusion map.

The question arise: the inclusion map seems to be similar to the boundary map.Would it be possible to construct a homology based on the inclusion map δkinstead of the boundary map ∂k ?

A trivial observation shows that it is possible if the field F has characteristic2. Indeed, in this case −1 = 1 and so the inclusion map is the same as theboundary map.

I shall look at the following example. Let Δ be a simplicial complex and σ bea three dimensional face {α0, α1, α2, α3} Let δ be the inclusion map. Then

δ(α0, α1, α2, α3) = α1α2α3 + α0α2α3 + α0α1α3 + α0α1α2

δ2(α0, α1, α2, α3) = δ(α1α2α3 + α0α2α3 + α0α1α3 + α0α1α2)

= α2α3 + α1α3 + α1α2 + α2α3 + α0α3 + α0α2

+α1α3 + α0α3 + α0α1 + α1α2 + α0α2 + α0α1

= 2[α0α1 + α0α2 + α0α3 + α1α2 + α1α3 + α2α3]

δ3(α0, α1, α2, α3) = δ(2[α0α1 + α0α2 + α0α3 + α1α2 + α1α3 + α2α3])

= 6[α0 + α1 + α2 + α3]

= 3![α0 + α1 + α2 + α3]

37

Thus in the field of characteristic 3, δ3(σ) = 0 , since 3! = 0 in such a field. Iwill state the idea more generally in the next lemma.

Lemma 4.2.2 Let σ be a simplex on a set of n vertices. Let δ be the inclusionmap and p be a positive integer. Also let τ be a (n− p) -face of σ . Then

δp(σ) = p!∑

τ , (4.2)

where the sum runs through all the (n − p) -faces. Thus, if p is a prime, thenin a field of characteristic p we have δp = 0

Proof. The number of (k − 1) -faces of any k -face is k , because there are kways to delete a vertex from an array of k vertices.

The inclusion map on σ , δ(σ) , is a sum of all (n−1) -faces and there are n termsin this sum. Similarly δ2(σ) is a sum of all the (n−2) -faces, and there (n)(n−1)terms in this sum because the number of (n− 2) -faces of any (n− 1)− face is(n− 1) . In this manner, δp(σ) is the sum of all the (n− p)− faces , τ , and thenumber of terms in the sum is (n)(n− 1)(n− 2) · · · (n− p− 1) . The number of(n− p) -faces of σ is given by the binomial coefficient,(

nn− p

)=

(n)(n− 1)(n− 2) · · · (n− p− 1)

p!

For p > 1 , the number of terms in the sum of δp(σ) is greater than the numberof (k− p) -faces of σ . This means that there must be more than one copy of the(n − p) -faces in the sum of δp(σ) . Thus we may write δp(σ) = t

∑τ , where t

is the number of copies of any (n− p) -face. So

t =(n)(n− 1)(n− 2) · · · (n− p− 1)(

nn− p

) = p! .

Hence, if characteristic of F is equal to p , we have δp = 0 . �

4.3 Modular Homology of Simplicial Complexes

In this Section I shall describe a variant of the generalized homology that isbased on inclusion maps and exists only over fields of finite characteristic. Thisis the modular homology first defined in [3] and [17]. The modular homologyis not homotopy invariant and so is not so useful in topology. However, it hasproperties (which we shall see) which are of interest in combinatorics.

38

Definition 4.3.1 Let F be a field of characteristic p > 2 and Δ be a simplicialcomplex on a finite set of cardinality n . Let MΔ

k denote the F -vector spacespanned by the k -faces of Δ , and let δ be the inclusion map. Then the sequence

MΔ : 0δ←MΔ

0δ←MΔ

1δ←MΔ

2δ← · · · δ←MΔ

k−1δ←MΔ

kδ← · · · (4.3)

is called the p-modular homological sequence for Δ .

As we have seen, δp = 0 and so (4.3) is the generalized homological sequence.In other words, for every j and 0 < i < p , a subsequence of (4.3) of the form

· · · δp−i← MΔj−i

δi←MΔj

δp−i← MΔj−i+p

δi← · · · (4.4)

is homological.

The sequence (4.4) is uniquely determined by any arrow MΔl ←MΔ

r in it, and soit is denoted by MΔ

(l,r) ( l and r mean left and right). The number of sequences

MΔ(l,r) , depends on the characteristic of the field. For example when p = 3 ,

there are three sequences of the form (4.4). In general there are(

p2

)= p2−p

2

ways to choose an arrow MΔl ←MΔ

r , so there are p2−p2

homological sequences.

Definition 4.3.2 The unique arrow MΔa ← MΔ

b in MΔ(l,r) , for which 0 ≤

a+ b ≤ p is called the initial arrow. The module MΔb is called the 0-position of

MΔ(l,r) and (a, b) is referred to as the type of MΔ

(l,r)

The position of any module in MΔ(l,r) is counted from the 0-position.

Definition 4.3.3 For any arrow MΔl ← MΔ

r the quantity (l + r) is calledthe weight of the arrow. Further, the weight of the sequence of MΔ

(l,r) is the

integer w with 0 < w ≤ p such that w ≡ l + r − n (mod p) .

Definition 4.3.4 Let MΔ be a p -modular sequence of inclusion maps asso-ciated with the simplicial complex Δ . Let

· · · ←MΔj−i

δi←MΔj

δp−i← MΔj−i+p ← · · ·

be a homological subsequence of MΔ . The homology at Mj given by

HΔj,i = (Ker δi ∩Mj)/(Im δp−i ∩Mj)

is called the p-modular homology. Dimension of HΔj,i is the p-modular Betti

number βΔj,i

39

Just as above, if MΔ(l,r) has the only one non-trivial homology, then MΔ

(l,r) is

called almost exact. If MΔ(l,r) is almost exact, then the non-trivial homology is

denoted HΔ(l,r) and dim(HΔ

(l,r)) = βΔ(l,r) , where βΔ

(l,r) is the only non-zero Betti

number of MΔ(l,r) .

Definition 4.3.5 It is said that βΔ(l,r) or HΔ

(l,r) is at position d if the non-

trivial homology occurs at position d (counted from the initial position). IfMΔ

(l,r) is exact or almost exact for every choice of l and r , then MΔ is almostp-exact

We shall now look at an example .

Example 4.3.6 Let F be GF(3) and let Δ be a simplicial complex on avertex set of cardinality 10. We could construct three homological sequences asfollows

· · · 0 0←M13←M2

6←M49←M5

12←M715←M8

18←M10 ← 0 · · · (4.5)

· · · 0 −1← M02←M2

5←M38←M5

11←M614←M8

17←M9 ← 0 · · · (4.6)

· · · 0 −2← M01←M1

4←M37←M4

10←M613←M7

16←M9 ← 0 · · · (4.7)

Note that in the above sequences the weight of each arrow is written on top ofthe corresponding arrow. The modules to the left of M0 are denoted M−1 , M−2

and so on. For this reason the leftmost arrow in (4.5) has a weight of 0, while in(4.6) it is -1 and in (4.7) it is -2.

4.4 Modular Homology of a Simplex Σn

In this section I shall describe the modular homology of a simplex Σn as it hasbeen done in [17, 19]. Instead of using Σn in the subscript, I will use n . Thus

MΣn

=: Mn, MΣn

(l,r) =: Mn(l,r), HΣn

(l,r) =: Hn(l,r), HΣn

j,i =: Hnj,i, etc.

We shall consider the sequence

Mn : 0δ←Mn

0δ←Mn

1δ←Mn

2δ← · · · δ←Mn

k−1δ←Mn

kδ← · · · (4.8)

Note that the above sequence is similar to (4.3), except that now Δ = Σn .

The next result has been proved in [17]:

40

Proposition 4.4.1 Let F be a field of characteristic p > 2 , and let Σn be asimplex with n vertices. For integers 0 < l < r ≤ p , the sequence Mn

(l,r) , iseither exact or almost exact.

Since there could be at most one non-trivial homology in Mn(l,r) , this non-trivial

homology is denoted as Hn(l,r) and the non-trivial Betti number (the dimension

of Hn(l,r) ) is denoted as βn

(l,r) . The position at which the non-trivial homologyoccurs is denoted as dn

(l,r) .

The following proposition gives a criterion to determine whether for a simplex,a given sequence Mn

(l,r) is exact or not.

Proposition 4.4.2 (The Middle-Term Condition)

The homology Hnj,i = 0 unless n− p < 2j − i < n .

The next result can be used to find the position of the non-trivial homology in asequence Mn

(l,r) :

Proposition 4.4.3 Let Mn(l,r) be a homological sequence with weight w. If

w = p , then the sequence Mn(l,r) is exact. However if 0 < w < p , then Mn

(l,r) isalmost exact and the non-trivial homology occurs at position

dn(l,r) =

⌊n− v

p

⌋,

where v is the weight of the initial arrow in Mn(l,r) .

The next example illustrates the previous proposition.

Example 4.4.4 Let us consider the homological sequences of example (4.3.6).I will begin by looking at the sequence (4.5). First let us find the weight, w ofthis sequence. Taking l = 1 and r = 2 , w ≡ 1+2−10(mod 3) = 2 . The weightof the initial arrow is 0. The previous proposition gives the position at which thenon-trivial homology occurs.

dn(l,r) =

⌊10 − 0

3

⌋= 3

Thus the non-trivial homology occurs at position 3. Position 3 corresponds tothe space M5 . (M1 is taken to be at position 0.) For the sequence (4.6), takingl = 0 and r = 2 , the weight of the sequence, w ≡ 0 + 2 − 10(mod 3) = 1 . Theweight of the initial arrow is 2. From the previous proposition, the position ofthe non-trivial homology,

41

dn(l,r) =

⌊10 − 2

3

⌋= 2

In this case the non-trivial homology occurs at position 2. Position 2 correspondsto the space M5 and in this sequence, M2 is at position 0. For the last sequence,(4.7), taking l = 0 and r = 1 , we have w ≡ 0 + 1 − 10(mod 3) = 0 . Since0 < w ≤ p , we w = p . Thus by the previous proposition, this sequence is exact.

�The value of the unique non-trivial Betti number, βn

(l,r) , can also be determined.The Euler-Poincare equation gives

∞∑k=−∞

dimMnl+kp − dimMn

r+kp = ±dimHn(l,r) = ±βn

(l,r)

and since Betti number is non-negative, we have

βn(l,r) =

∣∣∣∣∣∞∑

k=−∞dimMn

l+kp − dimMnr+kp

∣∣∣∣∣Note that the dimension of Mn

l+kp is given by the binomial coefficient

(n

l + kp

).

Hence

βn(l,r) =

∣∣∣∣∣∞∑

k=−∞

(n

l + kp

)−

(n

r + kp

)∣∣∣∣∣The next fact has been proven in [17]

Proposition 4.4.5 In the field of characteristic 3, βn(l,r) is either 0 or 1.

4.5 Group Actions on a Simplex

Let G be a permutation group of degree n acting on a set Ω . In other words,elements of G permute elements of a set Ω , |Ω| = n . Then there is a naturalaction of G on 2 -element subsets of Ω , 3 -element subsets of Ω and so on.

Of course, a set of cardinality n , together with all its subsets is the simplex Σn ,of dimension n− 1 . This simplex is also called the boolean algebra Bn .

Definition 4.5.1 Let F be a field and let Σn be a simplex on the vertex setΩ , |Ω| = n . Let Mn

k be the space spanned by the k -faces of Σn and also let G

42

be a permutation group of degree n . Let f = λ0σ0+λ1σ1+λ2σ2+ . . .+λjσj ∈Mn

k . Here σi is the k -face of Σn and λi ∈ F . Then the natural action of Gon f is given by

f g = λ0(σ0)g + λ1(σ1)

g + λ2(σ2)g + . . .+ λj(σj)

g (g ∈ G)

Note that j is the number of k -faces of Σn and is given by j =

(nk

).

So G acts on the different levels of the boolean algebra Bn . The orbits of G onthe set of k -element subsets of Ω , are called the symmetrized k -orbits of G . Iwill call them briefly k -orbits. Let Nk := NG

k be the number of k -orbits of thegroup G .

Example 4.5.2 Let F be a field. Let Ω = {α0, α1, α2, α3, α4} be a set with

five elements. Let G =⟨(

0 1 2 3 41 2 3 4 0

)⟩, the cyclic group Z5 . In this case

Mn1 is the 5-dimensional vector space over F . The natural basis for Mn

1 are thefive 1-subsets of Ω . Similarly Mn

2 is a 10-dimensional F -vector space spanned by

the 10 2-subsets of Ω . Let us consider the action of g =

(0 1 2 3 44 2 1 0 3

)∈ G

on f = {α0} − 2{α1} + {α4} ∈ Mn1 . (Here I am assuming that F has at least

three elements)f g = {α0}g − 2{α1}g + {α4}g

= {α4} − 2{α2} + {α3}

Definition 4.5.3 Let G be a group acting on a simplex Σn . An elementf ∈ Mn

k is called G-invariant if f g = f , for every g ∈ G . The set of allG-invariants form a subspace in Mn

k . This subspace is denoted by MGk

LetfG :=

∑g∈G

f g.

Note that fG is always G -invariant. Indeed, take any h ∈ G , then

(fG)h =∑g∈G

(f g)h =∑g∈G

f gh =∑g∈G

f g = fG

So as fG is the sum of all the elements in the orbit of f , we have the followingproposition:

43

Proposition 4.5.4 Let F = R . Let a group of degree n acts on a simplexΣn . A natural basis of MG

k is formed by the G -invariant elements of the form

(xi)G =

∑g∈G

(xi)g

where xi ∈Mnk . Furthermore, the dimension of MG

k is equal to Nk , the numberof k -orbits of G .

Example 4.5.5 Let F be a field, let Ω = {α0, α1, α2, α3, α4} be a set with

five elements and let G =⟨(

0 1 2 3 41 2 3 4 0

)⟩be the cyclic group Z5 . Let us

consider elements of spaces MG1 and MG

2 :

take f = {α0} + {α1} + {α2} + {α3} + {α4} ∈Mn1 then

f g = {α0}g + {α1}g + {α2}g + {α3}g + {α4}g = f, (g ∈ G)

Thus f g is an element of the space MG1 .

f2 = {α0, α1}+{α1, α2}+{α2, α3}+{α3, α4}+{α4, α0} ∈Mn2 is another example

of a G -invariant element since

f g2 = {α0, α1}g + {α1, α2}g + {α2, α3}g + {α3, α4}g + {α4, α0}g = f2 (g ∈ G)

4.6 The Sequence of Invariant Spaces

In this section I will use some of the results from the previous section to formulateresults about the number of k-orbits.

As stated by proposition (4.5.4), the number of k-orbits is equal to the dimensionof MG

k . Hence it will be useful to consider the sequence of invariant subspaces.

Let Ω{k} be the set of all k -subsets of Ω . Let x ∈ Ω{k} be any such k -subset.Let OrbG(x) = {xg : g ∈ G} be the corresponding k -orbit of G . As it has beennoticed above, the invariant subspaces are spanned by elements

x =∑

y∈OrbG(x)

y

Thus there is a 1 : 1 -correspondence between k -orbits OrbG(x) of G and thebasis element x ∈MG

k .

Thus, the space MGk is a subspace of Mn

k and so we may restrict the map δ onMG

k to get the sequence

· · · ←MGj−i ←MG

j ←MGj−i+p ← · · ·

44

It has been proved in [17] that this sequence is homological. So we may talkabout homologies

HGj,i := (Ker δi ∩MG

j )/δp−iMGj−i+p

The dimension dimHGj,i = βG

j,i .

The next result (also proved in [17]) will be important.

Lemma 4.6.1 Let F be a field of prime characteristic p , and let G be a groupsuch that |G| = 0 in F . Now suppose that for the sequence

· · · ←Mnj−i ←Mn

j ←Mnj−i+p ← · · ·

holds Hnj,i = 0 . Then for the sequence

· · · ←MGj−i ←MG

j ←MGj−i+p ← · · · (4.9)

also holds HGj,i = 0 .

Proof. Suppose that (4.9) is not exact. This means that for some x ∈ MGj

such that δix = 0 , it is impossible to find any y ∈MGj−i+p such that x = δp−i(y)

On the other hand,

x =∑

z∈OrbG(x)

z

is an element of Mnj such that δix = 0. Hence there is an element t ∈ Mj−i+p

such that δp−i(t) = x. If it would be possible to show that t ∈ Mj−i+p (that isthat t is a G -invariant element) then the contradiction would be found.

Take any element g ∈ G and note that δp−i(tg) = xg = x.

Thusδp−i(

∑g∈G

tg) = |G|x

and so

δp−i(1

|G|∑g∈G

tg) = x

Hence, if |G| = 0 in the field F , then for

y =1

|G|∑g∈G

tg

would x = δp−i(y). The contradiction proves the lemma. �

Corollary 4.6.2 If · · · ← Mnj−i ← Mn

j ← Mnj−i+p ← · · · is an almost exact

sequence, and |G| = 0 in F , then the sequence MGj−i ← MG

j ← MGj−i+p ← · · ·

is also almost exact.

45

I will now illustrate how the elements of MGj are mapped to elements of MG

j−i .

Example 4.6.3 Let Σn be the simplex on the vertex set Ω = {α0, α1, α2, α3, α4, α5}.Let G be the cyclic group Z6 =

⟨(0 1 2 3 4 51 2 3 4 5 0

)⟩and let F be a field. Let

Mnk be the F -vector space spanned by the k -subsets of Ω and let MG

k be thespace containing the invariant elements of Mn

k .

Let orbG(x) be the set of all elements in the same orbit as x . Then

x =∑

y∈orbG(x)

y

is a G -invariant element as shown in example (4.5.5). Thus x ∈MGk .

Now let us look at the 3-subset {α0, α2, α4} .

orbG({α0, α2, α4}) = {{α0, α2, α4}, {α1, α3, α5}}

so {α0, α2, α4} + {α1, α3, α5} is a G -invariant and thus an element of MG3 .

δ({α0, α2, α4} + {α1, α3, α5}) = δ{α0, α2, α4} + δ{α1, α3, α5}= {α0, α2} + {α0, α4} + {α2, α4}+

{α1, α3} + {α1, α5} + {α3, α5}= {α0, α2}

Thus we have seen an example of δ : MG3 →MG

2 .

Consider another element k = {α0, α1, α2} . So

k = ({α0, α1, α2}+{α1α2α3}+{α2α3α4}+{α3α4α5}+{α4α5α0}+{α5α0α1}) ∈MG3

δk = δ({α0, α1, α2}+{α1α2α3}+{α2α3α4}+{α3α4α5}+{α4α5α0}+{α5α0α1})= {α0, α1} + {α0, α2} + {α1, α2} + {α1, α2} + {α1, α3}+

{α2, α3} + {α2, α3} + {α2, α4} + {α3, α4} + {α3, α4}+{α3, α5} + {α4, α5} + {α4, α5} + {α4, α0} + {α5, α0}+{α5, α0} + {α5, α1} + {α0, α1}

= 2 [{α0 α1} + {α1, α2} + {α2, α3} + {α3, α4} + {α4, α5} + {α5, α0}] +[{α0, α2} + {α1, α3} + {α2, α4} + {α3, α5} + {α4, α0} + {α5, α1}]

= 2{α0, α1} + {α0, α2}So δ({α0, α1, α2}) = 2{α0, α1} + {α0, α2} �

46

Once again looking at the sequence

· · · ←MGj−i ←MG

j ←MGj−i+p ← · · · , (4.10)

we see that the Euler characteristic of this sequence is

±χ =+∞∑

k=−∞(dimMG

j−i+kp − dimMGj+kp) =

+∞∑k=−∞

(NGj−i+kp −NG

j+kp)

On the other hand

χ =+∞∑

k=−∞(βG

j−i+kp − βGj+kp)

From proposition (4.4.1) and ( 4.6.1) we know that the sequence (4.10) is almostexact. Thus we may write,

χ = ±βG(l,r)

where βG(l,r) is the unique non-zero Betti number. From this and keeping in mind

the fact that β ≥ 0 we get

βG(l,r) =

⏐⏐⏐⏐⏐+∞∑

k=−∞(NG

j−i+kp −NGj+kp)

⏐⏐⏐⏐⏐or

βG(l,r) =

⏐⏐· · · +NGj−i −NG

j +NGj−i+p −NG

j+p + · · ·⏐⏐ (4.11)

If G is trivial then the number of k-orbits is

(nk

)and in this case

βG(l,r) =

⏐⏐⏐⏐⏐+∞∑

k=−∞

(n

j − i+ kp

)−

(n

j + kp

)⏐⏐⏐⏐⏐Theorem 4.6.4 Let a group of an odd order acts on a simplex Σn . Let nis the degree of the group G and m = �n

2� . Then the 2-modular homological

sequence· · · ←MG

m−1 ←MGm ←MG

m+1 ←MGm+2 ← · · ·

is exact.

Proof. The theorem (4.6.4) follows from the fact that in GF(2) , −1 =1 . Hence the 2-modular homological sequence is the same as the simplicialhomological sequence and all the Betti numbers are equal to zero. �So for p = 2 , the equation (4.11) becomes

0 = · · · +Nm−2 −Nm−1 +Nm −Nm+1 +Nm+2 −Nm+3 + · · · (4.12)

47

(Note that Nk := NGk ). When the degree of G is odd, Nm−k = Nm+k+1 , and

all the terms of (4.12 cancel out and we do not have any information about thenumber of k-orbits. However when the degree of G is even, Nm−k = Nm+k andequation (4.12) could be written as

0 = Nm − 2[Nm−1 −Nm−2 +Nm−3 −Nm−4 + · · ·].

Thus, the previous result implies the following Corollary:

Corollary 4.6.5 Let a group of an odd order |G| and of an even degree n =2m acts on a simplex Σn . Let Nk be the number of orbits of G on k -faces ofΣn . Then the following equality holds:

Nm

2= Nm−1 −Nm−2 +Nm−3 −Nm−4 + · · · . (4.13)

I will illustrate the previous result:

Example 4.6.6 Let us look at the action of

G =

{(1 2 3 41 2 3 4

) (1 2 3 43 1 2 4

) (1 2 3 42 3 1 2

)}� Z3

on the set {α1, α2, α3, α4} . Let this action be defined by the rule gαi = αg(i) ,for g ∈ G and αi ∈ Ω .

From the previous corollary we have, N2 = 2[N1 −N0] . I shall try to verify thisequation.

The 2-subsets of Ω are

{{α1, α2}, {α1, α3}, {α1, α4}, {α2, α3}, {α2, α4}, {α3, α4}}

G has two orbits, namely,

orbG({α1, α2}) = {{α1, α2}, {α1, α3}, {α2, α3}}

andorbG({α1, α4}) = {{α1, α4}, {α3, α4}, {α2, α4}}

Thus N2 = 2 . There are also two orbits on 1-subsets and these are orbG({α1}) ={{α1}, {α2}, {α3}} and orbG({α4}) = {{α4}}, so N1 = 2 . N0 = 1 , since thenull space of Ω is the only 0-subset.

The equation N2 = 2[N1 −N0] indeed holds.

48

4.7 Improvement of Inequalities

This subsection contains my own results. In the paper [17], the following theoremgives inequalities for a group whose order is co-prime to 3.

Theorem 4.7.1 Let G be a group of order co-prime to 3 and of degree n . LetNk be the number of k -orbits of G and let N :=

∑nk=0Nk be the total number

of orbits.

(i) If n = 2m , then

N − 3Nm

6≤

∑k≥1

Nm−3k ≤ N −Nm

6

(ii) If n = 2m+ 1 , then

N − 2Nm

6≤

∑k≥0

Nm−1−3k ≤ N

6

Using the materials so far and especially proposition (4.4.5), I was able to improvethe above theorem as follows:

Theorem 4.7.2 Let G be a group of order co-prime to 3 and of degree n . Thenthe following inequalities for the numbers of k -orbits hold:

(i) If n = 2m , then

N − 3Nm

6≤

∑k≥1

Nm−3k ≤ N − 3Nm + 2

6

ii) If n = 2m+ 1 , then

N − 2

6≤

∑k≥0

Nm−3k−1 ≤ N

6

Proof. Let Ω be a set of cardinality n and let Σn be the simplex on Ω . LetG be a group of degree n whose order is not divisible by 3, that acts on Σn .The following homological sequences could be constructed.

· · · ←MGm−3 ←MG

m−2 ←Mnm ←MG

m+1 ←MGm+3 ← · · · (4.14)

· · · ←MGm−3 ←MG

m−1 ←Mnm ←MG

m+2 ←MGm+3 ← · · · (4.15)

· · · ←MGm−2 ←MG

m−1 ←MGm+1 ←MG

m+2 ←MGm+4 ← · · · (4.16)

49

Each of these 3-modular homological sequences could be either exact or almostexact, and if it is almost exact then by corollary (4.4.5), the non-trivial Bettinumber is equal to 1. I will denote this unique non-trivial Betti number byβG

(l,r) .

By the Euler-Poincare equation,

βG(l,r) =

⏐⏐⏐⏐⏐+∞∑

k=−∞(NG

j−i+kp −NGj+kp)

⏐⏐⏐⏐⏐From this, let us first consider the case when the degree n = 2m is even andfind the expressions for βG

(l,r) . From the sequence (4.14),

βG(l,r) = Nm −Nm+1 −Nm−2 +Nm+3 +Nm−3 − · · ·

In this case Nm−k = Nm+k . Thus

βG(l,r) = Nm −Nm−1 −Nm−2 + 2Nm−3

−Nm−4 −Nm−5 + 2Nm−6

−Nm−7 −Nm−8 + 2Nm−9 · · ·= 2Nm − (Nm +Nm−1 +Nm−2 + · · ·)

+3(Nm−3 +Nm−6 +Nm−9 + · · ·)

For the sequence (4.15),

βG(l,r) = Nm −Nm−1 −Nm+2 +Nm−3 +Nm+3 − · · ·

= Nm −Nm−1 −Nm−2 + 2Nm−3

−Nm−4 −Nm−5 + 2Nm−6

−Nm−7 −Nm−8 + 2Nm−9 · · ·= 2Nm − (Nm +Nm−1 +Nm−2 + · · ·)

+3(Nm−3 +Nm−6 +Nm−9 + · · ·)Thus the Betti number of the sequence (4.15) is equal to the Betti number ofthe sequence (4.14)

For the sequence (4.16),

βG(l,r) = Nm+1 −Nm−1 −Nm+2 +Nm−2 +Nm+4 − · · ·

= Nm−1 −Nm−1 −Nm+2 +Nm−2 + · · ·= 0

50

Thus there are two almost exact and one exact sequence.

If we put N :=∑

k≥0Nk , then

N = Nm +Nm−1 +Nm+1 +Nm−2 +Nm+2 + · · ·= Nm + 2Nm−1 + 2Nm−2 + 2Nm−3 + · · ·

= 2(Nm +Nm−1 +Nm−2 +Nm−3 + · · ·) −Nm

andN +Nm

2= Nm +Nm−1 +Nm−2 +Nm−3 + · · ·

So for sequences (4.14) and (4.15), when n = 2m , βG(l,r) could be written as

βG(l,r) = 2Nm − N +Nm

2+ 3

∑k≥1

Nm−3k

=3Nm −N

2+ 3

∑k≥1

Nm−3k

From proposition (4.4.5), 0 ≤ βG(l,r) ≤ 1 . So we could write

0 ≤ 3Nm −N

2+ 3

∑k≥1

Nm−3k ≤ 1

orN − 3Nm

6≤

∑k≥1

Nm−3k ≤ N − 3Nm + 2

6(4.17)

This proves the first part of my proposed result.

For the case when the degree of G is odd, that is n = 2m+1 , Nm−k = Nm+k+1 .Using the Euler-Poincare equation again, we can get the expressions for the Bettinumber βG

(l,r) . For the sequence (4.14),

βG(l,r) = Nm −Nm+1 −Nm−2 +Nm−3 +Nm+3 − · · · = 0

For the sequence (4.15),

βG(l,r) = Nm −Nm−1 −Nm+2 +Nm−3 +Nm+3 − · · ·

= Nm − 2Nm−1 +Nm−2 +Nm−3

−2Nm−4 +Nm−5 +Nm−6

−2Nm−7 +Nm−8 +Nm−9 · · ·= [Nm +Nm−1 +Nm−2 +Nm−3 · · ·]

51

−3[Nm−1 +Nm−4 +Nm−7 · · ·]For the sequence (4.16),

βG(l,r) = Nm+1 −Nm−1 −Nm+2 +Nm−2 +Nm+4 − · · ·

= Nm − 2Nm−1 +Nm−2 +Nm−3

−2Nm−4 +Nm−5 +Nm−6

−2Nm−7 +Nm−8 +Nm−9 · · ·= [Nm +Nm−1 +Nm−2 +Nm−3 · · ·]

−3[Nm−1 +Nm−4 +Nm−7 · · ·]Thus again it occurs that there are two almost exact and one exact sequence andthe non-trivial Betti numbers in the sequences(4.15) and (4.16) are essentiallythe same. Let N :=

∑k≥1Nk . Since Nm−k = Nm+k+1 , we can write, N =

2Nm + 2Nm−1 + 2Nm−2 + · · · . The betti number for the sequence (4.15) and(4.16) could now be written as

βG(l,r) =

N

2− 3

∑k≥0

Nm−3k−1

Once again letting N :=∑

k≥1Nk and using corollary (4.4.5), we get the in-equality,

0 ≤ N

2− 3

∑k≥0

Nm−3k−1 ≤ 1

orN − 2

6≤

∑k≥0

Nm−3k−1 ≤ N

6(4.18)

�

4.8 Group Actions on Modular Homology

Other results about numbers of orbits are related with group actions on themodular homology. I shall describe now some of the facts about such actions.

Let Δ be a (n − 1) -dimensional simplex Σn . Let G be a group that acts onthe set Ω of vertices of Σn . As we have seen, then G acts also on the spacesMn

j .

52

I will show now that then G acts on the modular homologies Hnj,i also. Indeed,

elements of Hnj,i are cosets [x] := x+ δp−iMn

j+p−i , where x ∈Mnj and δix = 0 .

Define a G -action on the cosets by the rule

[x]g = [xg] = xg + δp−iMnj+p−i.

Then G acts on Hnj,i , and we may look at the orbits of G on Hn

j,i and at thespace (Hn

j,i)G of G-invariant elements of Hn

j,i . The next result has been provenin [17]:

Lemma 4.8.1 Let G be a permutation group of degree n acts on a simplex Σn

and let p be a prime such that |G| is not divisible by p . Let HGj,i denote the

homologies of the sequence

· · · ←MGj−i ←MG

j ←MGj−i+kp ← · · ·

Then HGj,i = (Hn

j,i)G

Corollary 4.8.2 Let F be a field of prime characteristic. Let a group G ,such that charF is not divisible by |G| , acts on a simplex Σn . Then

βGj,i ≤ βn

j,i

Proof. Indeed as the number of orbits on a set is no more than the numberof elements in the set, we have

βGj,i = dimHG

j,i = dim(Hnj,i)

G ≤ dimHnj,i = βn

j,i

�As we know, 3-modular Betti numbers of the sequence Mn are either 0 or 1.Then the previous result implies the following:

Corollary 4.8.3 Let G be a group such that the order of G is not divisibleby 3. Let

· · · ←MGj−i ←MG

j ←MGj+3−i ← · · · (4.19)

be an almost exact homological subsequence of a 3-modular homological sequenceMG . Then (4.19) is either exact or almost exact with the unique non-trivialBetti number equal to 1.

Sometimes this can be used to get some information about numbers of k -orbits.The next simple result belongs to me:

53

Proposition 4.8.4 If G is a 4-homogeneous group of degree 12 such that theorder of G is not divisible by 3, then N5 = N6 .

Proof. By (4.11),

βG(l,r) = N6 −N5 −N4 + 2N3 −N2 −N1 + 2N0.

If G is 4-homogenous, but not 5-homogenous, that is N0 = N1 = N2 = N3 =N4 = 1 and N5 ≥ 2 , then

βG(l,r) = N6 −N5 − 1 + 2 − 1 − 1 + 2

= N6 −N5 + 1

By the previous Corollary, βG(l,r) ≤ 1 . So we have

N6 −N5 + 1 ≤ 1 or N6 ≤ N5.

By the theorem of Livingstone and Wagner, there should be N6 ≥ N5 . Thismeans that N6 = N5. �

4.9 Generalized Livingstone and Wagner In-

equalities

We shall now consider the sequence

· · · ←Mnj−i

δ←Mnj

δ←Mnj−i+p ← · · ·

There is an action of G on each of the spaces Mnj and so there is a natu-

ral permutation representation of G on the FG -module Mnj . There is also a

representation of G on the modular homologies Hnj,i .

The relationship between the group actions on Mnj and the group actions on

Hnj,i is given by the modified version of the Hopf-Lefschetz formula∑

k≥0

tr(g,Mnj+kp)−tr(g,Mn

j−i+kp) =∑k≥0

tr(g,Hnj+kp)−tr(g,Hn

j−i+kp), (g ∈ G).

Since we are talking about modular homology of a simplex the formula abovecould be simplified to

±(∑k≥0

tr(g,Mnj+kp) − tr(g,Mn

j−i+kp)) = tr(g,Hnj,i), (g ∈ G) (4.20)

54

Here Hnj,i is the unique non-trivial homology.

On the other hand,

tr(g,Mnj ) =

∑U

cj(U)tr(g, U) (g ∈ G)

Here U is any irreducible representation of G over F , and cj(U) is the multi-plicity of U in the representation of G on Mn

j . The summation runs over allthe irreducible representations U .

Thus the left hand side of (4.20) could be written as

±+∞∑

k=−∞((∑

U

cj+kp(U)tr(g, U)) − (∑

U

cj−i+kp(U)tr(g, U)))

= ±+∞∑

k=−∞(∑

U

cj+kp(U)tr(g, U) − cj−i+kp(U)tr(g, U))

= ±∑

U

(+∞∑

k=−∞cj+kp(U) − cj−i+kp(U))tr(g, U) , (g ∈ G).

It follows from the previous formula and from (4.20), that

±+∞∑

k=−∞cj+kp(U) − cj−i+kp(U)

is equal to the multiplicity of U in the representation of G on the unique non-trivial homology Hn

j,i . Since multiplicities cannot be negative, we have

±+∞∑

k=−∞cj+kp(U) − cj−i+kp(U) ≥ 0 . (4.21)

For any fixed j and p , we could abbreviate the sums of the form

+∞∑k=0

cj−kp(U) or+∞∑

k=−∞cj−kp(U)

by (cj(U))p or [cj(U)]p respectively. Now (4.21)could be written as

±[cj(U) − cj−i(U)]p ≥ 0 (4.22)

where the sign ± depends on n , j and i only and is related with the positionof the nontrivial homology Hn

j,i . The inequality holds for the multiplicity of anyirreducible representation U .

55

Now let us consider the homological sequence

· · · ←Mnm−p+1 ←Mn

m ←Mnm+1 ← (4.23)

(here m := �n2� ). The middle term condition shows that the homology Hn

m,p−1

is non-trivial. Hence[cm(U) − cm−p+1(U)]p ≥ 0

or[cm(U)]p ≥ [cm−p+1(U)]p (4.24)

I will write [ck(U)]p as [ck]p and (ck(U))p as (ck)p. Note that trivial homologiesin the sequence (4.23) would give equations instead of inequalities. For this reasonwe are choosing non-trivial homologies.

Note that since [cm−p+1]p = [cm+1]p and [cm+1]p = [cm−1]p by the Livingstone-Wagner and Stanley theorems, the inequality (4.24) could be written as

[cm]p ≥ [cm−1]p

We could also show that [cm+1]p = [cm−1]p by looking at the trivial homologyHn

m+1,2 (result from the middle term condition) we would get

[cm+1]p − [cm−1]p = 0

Hnm−1,p−3 is non-trivial and using (4.24) we get as the result

[cm−1]p ≥ [cm−1−p+3]p = [cm+2]p.

Again, Hnm+2,4 is trivial and it gives [cm+2]p = [cm−2]p . Furthermore, Hn

m−2,p−5

is non-trivial and from (4.24) we have

[cm−2]p ≥ [cm−2−p+5]p = [cm+3]p.

Let s = p−12

. By choosing non-trivial homologies

Hnm+3,6, H

nm−3,p−7, · · ·Hn

m−s+1,2, Hnm+s,p−1

and combining all previous inequalities we get

[cm]p ≥ [cm−1]p ≥ [cm−2]p ≥ · · · ≥ [cm−s]p (4.25)

The above inequalities (4.25) may be simplified. To do this we consider the sum

[cm]p =+∞∑

k=−∞cm−kp = cm + cm−p + cm−2p + · · · + cm+p + cm−2p + · · ·

56

Now, for a simplex we have cm+p = cn−m−p . So

[cm]p = cm + cm−p + cm−2p + · · · + cm+p + cm+2p + cm+3p + · · ·

= cm + cm−p + cm−2p + · · · + cn−m−p + cn−m−p−p + cn−m−p−2p + · · ·= (cm)p + (cn−m−p)p

= (cm + cn−m−p)p.

Thus I have just written down the proof of the following theorem (first provedin [23]):

Theorem 4.9.1 Let G be a permutation group on a finite set Ω of cardinalityn. Let F be a field of finite characteristic p = 2s + 1 > 2 that does not dividethe order of G. Let U be any irreducible representation of G and let ck be itsmultiplicity in the permutation representation of G on k-subsets of Ω . Then theinequalities hold:

(cm + cr)p ≥ (cm−1 + cr+1)p ≥ · · · ≥ (cm−s−1 + cr+s+1)p.

where m := �n2� and r := n− p−m.

(Note that m := �n2� means that either n = 2m or n = 2m− 1

Since the multiplicity of the trivial representation on Mnk (i.e., U = 1 ) is equal

to the number of k-orbits of G , Theorem (4.9.1) implies the following result onNG

k :

Corollary 4.9.2 Let G be a permutation group on a finite set Ω of cardinalityn. Let F be a field of finite characteristic p = 2s + 1 > 2 that does not dividethe order of G. Then the inequalities hold:

(Nm +Nr)p ≥ (Nm−1 +Nr+1)p ≥ · · · ≥ (Nm−s−1 +Nr+s+1)p.

where m := �n2� and r := n− p−m.

For m < p ≤ n the inequalities of theorem (4.9.2) can be written as

Nn−p+1 ≥ Nn−p +N0 ≥ · · · ≥ Nm−s +Nm−p+s ≥ 2Nm−s−1

For p > n , the inequalities of theorem (4.9.2) are exactly the inequalities ofWagner and Livingstone.

In the next chapter I would be looking at some of the applications of theorem(4.9.2).

57

4.10 Modular Homology of q -Simplex

A generalization of the notion of simplicial complex has been done by Gian-CarloRota in 1971, who wrote in [25]:

A combinatorial generalization of the notion of simplicial complex isobtained by replacing the word “subset” by “subspace of a vectorspace” throughout. To ensure finiteness, we take here a vector spaceV of finite dimension over a finite field GF (q) , and define a q -complex as a family Q of subspaces of V , such that if W ∈ Q andM ⊆ W , then M ∈ Q .

A q -complex is simply an order ideal in the lattice L(V ) of subspacesof V , ordered by inclusion, and this should make the analogy withsimplicial complexes apparent.

Previously, we looked at modular homology of a simplex. In this section, I wouldconsider the more general modular homology of a q -simplex. When q = 1 , aq -simplex is precisely the simplex Σn described earlier.

The q -simplex and the q -simplicial complex are defined as follows

Definition 4.10.1 Let V (n, q) be a n-dimensional vector space over GF(q) .Let Δ be a set of subspaces of V (n, q) such that if τ ∈ Δ , and σ is a subspaceof τ then σ ∈ Δ . Under this condition, Δ is a q-simplicial complex.

Definition 4.10.2 Let Δ be a q -simplicial complex on the vector space V (n, q) .If Δ contains all the subspaces of V (n, q) then Δ is called a q -simplex and isdenoted by Σn(q)

In this section I would concentrate mainly on the modular homology of q -simplexas it has been constructed in [21]. It is a partial case of the more general modularhomology of q -simplicial complex.

Definition 4.10.3 Let F be a field and let V (n, q) be a n-dimensional vectorspace over GF(q) . Then Mn

k (q) is the F -vector space spanned by all the k-dimensional subspaces of V (n, q) . Furthermore Mn

k (q) = 0 for k < 0 andk > n .

For further discussions in this section I would denote Mnk (q) by Mn

k as thiswould not create any confusion.

Note that Mn0 is the one-dimensional vector space with the null space as the

basis.

58

The inclusion map δk : Mnk →Mn

k−1 defined by δ(X) :=∑Y for X ∈Mn

k andY ∈Mn

k−1 is a maximal proper subspace of X.

Thus we obtain a chain of inclusion maps

Mn(q) : 0 ←Mn0 ←Mn

1 ← · · · ←Mnk−1 ←Mn

k ← · · ·As we shall see, if the field F of characteristic p > 0 , does not divide q , thenδπ = 0 , for some π . Hence this gives rise to π -modular homologies.

Indeed, if we fix some i ≤ k and let X ∈Mnk , then

δi(X) = c∑

Y

where the sum runs over all Y ∈Mnk−i with Y ⊂ X and where c is the number

of saturated chains Y = Y0 ⊂ Y1 ⊂ . . . ⊂ Yi = X It is easy to see that

c = (i!)q := (1 + q)(1 + q + q2) . . . (1 + q + q2 + . . .+ qi−1).

If p divides q then δi(X) =∑Y = 0 for any i ≤ k and this case will be of no

importance to us. On the other hand, if p does not divide q , then there will bevalues of i with (i!)q ≡ 0 mod p .

Definition 4.10.4 For co-prime integers p and q let π(p, q) > 0 be the leastinteger π for which [π]q ≡ 0mod p .

If q ≡ 1 mod p then clearly π(p, q) = p while if p does not divide (q− 1) thenπ(p, q) < p . In either case qπ ≡ 1 mod p and if π ≥ 2 then ((π − 1)!)q = 0 modp while (π!)q ≡ 0 mod p.

Therefore δπ is the least power of δ such that δπ = 0 . In the case of simplexstudied previously, q = 1 , and we have π(p, 1) = p and δp = 0 .

Example 4.10.5 Let us find some values of π(p, q) . Let q = 2 and p = 3 .

π 1 2 3 4[π]2 1 3 7 15

[π]2mod 3 1 0 1 0

The smallest integer π for which [π]2mod3 ≡ 0 is 2. So π(3, 2) = 2

Next let q = 2 and p = 5

π 1 2 3 4 5[π]2 1 3 7 15 31

[π]2mod 5 1 3 2 0 1

59

Here π(5, 2) = 4

The table below lists some values for π(p, q) .

p\q 2 3 4 5 7 8 9 11 13 16 17 19 23 25 272 - 2 - 2 2 - 2 2 2 - 2 2 2 2 23 2 - 3 2 2 2 - 2 3 3 2 3 2 3 -5 4 4 2 - 4 4 2 5 4 5 4 4 2 - 47 3 6 3 6 - 7 3 3 2 3 6 6 3 3 211 10 5 5 5 10 10 5 - 10 5 10 10 11 5 513 12 3 6 4 12 4 3 12 - 3 6 12 6 2 1317 8 16 4 16 16 8 8 16 4 2 - 8 16 8 1619 18 18 9 9 3 6 9 3 18 9 9 - 9 9 6

Definition 4.10.6 Let F be a field of prime characteristic p > 2 and let Σn(q)be a simplex on V (n, q) where p does not divide q . Let Mn

j denote the F -vector space spanned by all the k -dimensional subspaces of Vn(q) . The sequence

Mn(q) : 0 ←Mn0 (q) ←Mn

1 (q) ← · · · ←Mnk−1(q) ←Mn

k (q) ← · · ·is called the π -modular homological sequence.

Now all basic notions of the modular homology could be defined just as for usualsimplicial complexes. For example, a subsequence of the form

· · · ←Mnj−i(q)

δi←Mnj (q)

δπ−i← Mnj−i+π(q) ← · · · (4.26)

is homological and its homology at Mnj (q) is the π -modular homology.

The theory of modular homology of a q -simplex has many similarities with themodular homology of a simplex since a q -simplex is a more general notion of asimplex. The Proposition(4.4.1) could be written for q -simplex as follows

Proposition 4.10.7 The π -modular homological sequence Mn(l,r)(q) is either

exact or almost exact

The middle term condition below tells us when the homology Hnj,i(q) in the

sequence Mn(l,r)(q) is non-trivial.

Theorem 4.10.8 (The Middle Term Condition)

Let F be a field of characteristic p > 0 and let Σn(q) be a q -simplex, wherep does not divide q. Let i < π := π(p, q) and j be positive integers. ThenHn

j,i(q) = 0 unless n− π < 2j − i < n+ π

60

Since there is at most one non-trivial homology in Mn(l,r)(q) , this homology is

denoted as Hnl,r(q) (instead of Hn

j,i(q) ) and its dimension is denoted by βnl,r(q) .

It has been proved in [23] that the results about orbits could be formulated forq -simplices also. The following theorem is an analog of theorem (4.9.2), statedfor the action of linear groups, G < GLn(q) , on the subspace lattice of Vn(q) .

Theorem 4.10.9 Let V (n, q) be a vector space over GF(q) and let F be afield of prime characteristic p , not dividing q .Let Mk

n be the F -vector spacespanned by all the k -dimensional subspaces of V (n, q) . Also let G be a subgroupof GL(n, q) , whose order is co-prime to p . Then the inequalities hold.

(Nm +Nr)π ≥ (Nm−1 +Nr+1)π ≥ · · · ≥ (Nm−s−1 +Nr+s+1)π.

where m = �n2� , r := n− p−m and s := �π

2�

61

Chapter 5

Numbers of k -Orbits ofPermutation Groups

The number of orbits of a permutation group of degree n acting on a finite setof cardinality n , can be found using the Burnside’s Lemma. To find numberof k -orbits, we can use the theorem of Polya and to use the theorem of Polya,we need to know the cycle index of the group. Unfortunately, for some groupsit is not always easy to find the cycle index. The method of modular homologycould be useful since it gives a good estimate of the number of orbits, withoutrequiring the cycle index of the group.

In this chapter I will use the method of modular homology to obtain estimates ofthe number of k -orbits when a permutation group of degree n acts on k -facesof a simplex (the collection of all k -subsets of a finite set). We shall look atsituations where the inequalities of corollary (4.9.2), for the number of k -orbitsof the group, could produce some results. These inequalities arise when the orderof the group is not divisible by certain primes (as stated in corollary (4.9.2)). Theresults obtained from these inequalities are non-trivial if the order of the groupis sufficiently large. In the next section, I will explain the meaning of non-trivialand state exactly how large the order of the group needs to be in order to givenon-trivial results. The main focus this chapter is to look at applications ofcorollary (4.9.2) stated in chapter 3.

The results obtained in this chapter are general results. This means that therecould be several groups for which the results hold. I shall however take somefamiliar groups as examples where ever possible. As stated previously, the setof all subsets of a finite set of cardinality n (the Boolean algebra Bn ), and asimplex Σn , on n vertices have the same meaning in this thesis. Thus the wordsk -subsets and k -faces have been used to mean the same.

62

5.1 Conditions For Non-triviality

The following results are simple to show

Proposition 5.1.1 When a permutation group of degree n acts on the k -facesof a simplex Σn or equivalently on k -subsets of a finite set of cardinality n , thefollowing inequality holds: (

nk

)|G| ≤ Nk(G) ≤

(nk

)

(Here(

nk

)is the number of k -subsets of a set of cardinality n )

Proof. The number of k -subsets is equal to(

nk

)and the number of orbits

cannot exceed the number of elements in a set thusNk ≤(

nk

).

On the other hand, the length of the largest orbit can lot exceed |G| . Suppose

that all the orbits are of the length |G| , then there are(

nk

)/|G| orbits and

this is the least number of orbits that can exist. Thus

Nk ≥(

nk

)|G|

�

Proposition 5.1.2 Let a permutation group of degree n acts on the k -faces of

the simplex Σn . If(

nk

)does not divide |G| , then Nk(G) ≥ 2

Proof. Let us assume that there is only one orbit of G on the set of all k -

faces. This means that |orbG(x)| =(

nk

), where x is any k -face. Let Gx be

the stabilizer of x in G . The Burnside’s Lemma states that

|G||orbG(x)| = Gx.

In other words, the number of elements in an orbit divides the order of the group.

In our case, since there is only orbit of G on the k -faces,(

nk

)divides |G| .

Thus, we have a contradiction. �Let a permutation group G of degree n acts on the k -faces of the simples Σn .For an integer M ≥ 1 , the inequality Nk ≥M will be called trivial if

|G| ≤(

nk

)M

63

and non-trivial if

|G| >(

nk

)M

.

This criteria for results to be trivial or non-trivial follows directly from Proposi-tion 5.1.1. In the case when M = 2 , the additional condition for non-triviality

is that(

nk

)must divide the order of the group as stated in the proposition

(5.1.2). So G has sufficiently large order (meaning that it could produce non-

trivial results) if |G| >(

nk

)M

.

5.2 Inequalities for Numbers of Orbits

In this section I shall use Theorem 4.9.2 to get the inequalities for numbers ofk -orbits of permutation groups. Since the focus is on Theorem 4.9.2 I will restateit here.

Theorem 5.2.1 Let G be a permutation group on a finite set Ω of cardinalityn. Let F be a field of finite characteristic p = 2s + 1 > 2 that does not dividethe order of G. Then the inequalities hold:

(Nm +Nr)p ≥ (Nm−1 +Nr+1)p ≥ · · · ≥ (Nm−s−1 +Nr+s+1)p.

where m := �n2� and r := n− p−m.

For an action of a permutation group G of degree n on the simplex Σn , I willbe writing k -orbit of G to mean orbits of G as it acts on the k -faces of Σn .

Let us begin by looking at a corollary of the previous result.

Let G be a permutation group of degree 24 and suppose that the order of Gis not divisible by the primes 13, 17 and 19. I will use Theorem 4.9.2 to getinequalities for numbers of k -orbits of G . (n , the degree of G is 24 andm = �n

2� = 12 ). So we get the following inequalities:

For p = 13, N12 ≥ N11 +N0 ≥ N10 +N1 ≥ N9 +N2 ≥ ... ≥ N6 +N5 ≥ 2N5

For p = 17, N8 ≥ N7 +N0 ≥ N6 +N1 ≥ N5 +N2 ≥ N4 +N3 ≥ 2N3

For p = 19, N6 ≥ N5 +N0 ≥ N4 +N1 ≥ N3 +N2 ≥ 2N2

Solving the system yields the following result.

N6 ≥ 2, N8 ≥ 3, N12 ≥ 4

As stated in Section 5.1, the non-triviality of an inequality depends on the orderof the group G . Thus I will state size of the group or the restriction on the orderof group, for the results to be non-trivial.

64

• N6 ≥ 2 is non-trivial if |G| >(

246

)2

= 67, 298 and |G| is divisible by(246

).

• N8 ≥ 3 is non-trivial if |G| >(

248

)3

= 245, 157 .

• N12 ≥ 4 is non-trivial if |G| >(

2412

)4

= 676, 039 .

An example of a group for which the results N8 ≥ 3 and N12 ≥ 4 hold, is thesporadic simple Mathieu group M24 . (The order of M24 is 244,823,040 and isnot divisible by 13, 17 and 19. The result N6 ≥ 2 , is however trivial in the caseof M24 , since |M24| is not divisible by

(246

).)

I have done similar computations for several groups and I have summarized theresults in the table below.

As mentioned previously, the results in the table hold for any group which satisfiesthe given requirements. I have taken some famous groups from the Atlas [35]as examples. In some cases the groups that I have picked as examples fail tohave sufficiently large order. However these groups do have the correct degreeand do not have the required primes in their prime decomposition.

65

Let G be a permutation group of degree n act on the simplex Σn . In the tablebelow I have taken different values of n and assumed that |G| is not divisibleby certain prime(s). I was able to get the lower bound of the number of k -orbitsof G .

|G| not Example Results Conditions for commentn divisible by non-triviality

10 7 A6 N2 ≥ 2 45||G| holds forand A5 A6

11 7 M11 N5 ≥ 2 462||G|and L2(11)

12 7 M12, L2(11) N6 ≥ 2 924||G|and M11

13 7 and 11 L3(3) N3 ≥ 2 286||G|

N7 ≥ 3 |G| > 572 holds forL3(3)

14 11 L2(3) N4 ≥ 2 1, 001||G|

14 11 and 13 N2 ≥ 2 91||G|23.L3(2)

and L2(7)N4 ≥ 3 |G| > 333 holds for

23.L3(2)

15 11 and 13 A8 N3 ≥ 2 455||G|and A6

N5 ≥ 3 |G| ≥ 1, 001 holds forA8

66

|G| is not Examples Results Conditions for commentn divisible by non-triviality

16 11 and 13 M20 N4 ≥ 2 1, 820||G|

N6 ≥ 3 |G| > 2, 669

18 11 and 13 L2(17) N6 ≥ 2 18, 564||G|

N8 ≥ 3 |G| > 14, 586

18 11, 13 and 17 A6 N2 ≥ 2 153||G|

N6 ≥ 4 |G| > 4, 641

N8 ≥ 5 |G| > 8, 751

20 11, 13, 17 M20 and L2(19) N2 ≥ 2 190||G|and 19

N4 ≥ 3 |G| > 1, 615

N8 ≥ 5 |G| > 25, 194

21 13, 17 and 19 L2(7) N3 ≥ 2 1, 330||G|

N5 ≥ 3 |G| > 6, 783

N9 ≥ 5 |G| > 58, 786

67

|G| is not Examples Results Conditions for commentn divisible by non-triviality

22 13,17 and 19 M22 N4 ≥ 2 7, 315||G| to be

N6 ≥ 3 |G| > 24, 871 holds for M22

N10 ≥ 4 |G| > 161, 662 holds for M22

23 13,17 and 19 M23 N5 ≥ 2 33, 649||G| to be

N7 ≥ 3 |G| > 81, 719 holds for M23

N11 ≥ 4 |G| > 338, 020 holds for M23

24 13, 17, and 19 M24, A5, N6 ≥ 2 134, 596||G|L2(23)and M20

N8 ≥ 3 |G| > 254, 157

N12 ≥ 4 |G| > 676, 039

26 17, 19 and 23 L2(27) N4 ≥ 2 14, 950||G|

N8 ≥ 3 |G| > 520, 758

N10 ≥ 4 |G| > 1, 327, 933

28 17, 19 and 23 U3(3), L2(8), N6 ≥ 2 376, 740||G|L3(2)and S6(2)

N10 ≥ 3 |G| > 4, 374, 370

N12 ≥ 4 |G| > 7, 605, 438

68

|G| is not Examples Results Conditions for commentn divisible by non-triviality

30 17, 19 and 23 L2(29) N8 ≥ 2 5, 852, 925||G|

N12 ≥ 3 |G| > 28, 831, 075

N14 ≥ 4 |G| > 36, 355, 668

30 17, 19, 23 24.A8 N4 ≥ 2 435||G|and 29

N8 ≥ 4 |G| > 1, 463, 231

N12 ≥ 6 |G| > 14, 415, 537

N14 ≥ 7 |G| > 20, 774, 667

31 17, 19, 23 L5(2) N3 ≥ 2 4, 495||G|and 29 and L3(5)

holds forN9 ≥ 3 |G| > 6, 720, 025 L5(2)

N13 ≥ 4 |G| > 51, 563, 268

N15 ≥ 5 |G| > 60, 108, 039

32 17, 19, 23 L2(31) N4 ≥ 2 35, 960||G|and 29

N10 ≥ 4 |G| > 16, 128, 060

N14 ≥ 5 |G| > 94, 287, 120

N16 ≥ 6 |G| > 100, 180, 065

69

|G| is not Examples Results Conditions for commentn divisible by non-triviality

33 19, 23 and 29 L2(33) N5 ≥ 2 237, 336||G|

N11 ≥ 4 |G| > 48, 384, 180

N15 ≥ 5 |G| > 259, 289, 580

36 19, 23, 29 U3(3), L2(8) N6 ≥ 2 1, 947, 792||G|and 31 and S6(2)

N8 ≥ 3 |G| > 10, 086, 780

N14 > 4 |G| > 949, 074, 300

N18 > 6 |G| > 1, 512, 522, 550

40 23, 29, 31 A5 N4 ≥ 2 91, 390||G|and 37

N10 ≥ 4 |G| > 211, 915, 132

N12 ≥ 5 |G| > 1, 117, 370, 696

N17 ≥ 6 |G| > 14, 788, 729, 800

N18 ≥ 7 |G| > 161, 971, 802, 600

70

Notations of groups used

Group Abbreviation

Alternating groups An

Mathieu groups M11, M12, M22, M23, M24, M20 = 24 : A5

Symplectic group S6(2)

Unitary group U3(3)

Non-split extension 23 · L3(2)

Linear groups Ln(q)

Every row in the tables above gives information about a set of groups and thisinformation could also be written in forms of statements. I will state here twosuch statements

1. Let G be a permutation group of degree 22, such that the order of Gis sufficiently large and not divisible by 13, 17 and 19. Then G is not4-transitive and there are at least 3 orbits on 6-subsets and 4 orbits on10-subsets.

2. Let H be a permutation group of degree 10 such that |G| is a multiple of45 and not divisible by 7. Then H cannot be 2-transitive.

5.3 Some General Results

After looking at results obtained for the number of k -orbits various groups, Ihave been able to state the following general results:

Proposition 5.3.1 Let G be a group of degree n and let m := �n/2� . Letk be the number of primes p ∈ Z that do not divide the order of G and satisfythe inequality m < p ≤ n . Then the group G has at least k + 1 orbits onm -subsets, i.e.,

Nm ≥ k + 1 .

71

Proof. Let us suppose that the set {p1, p2, · · · , pk} of primes, satisfying theconditions of the proposition, is arranged in ascending order. Thus p1 corre-sponds to the smallest prime while pk corresponds to the largest prime. By thetheorem above, each prime will generate inequalities of the form

Nn−p+1 ≥ Nn−p +N0 ≥ Nn−p−1 +N1 · · · ≥ 2

So we will have k rows of inequalities corresponding to the k primes in the set{p1, p2, · · · , pk} .

For p = p1 Na ≥ Na−1 + N0 ≥ Na−2 + N1 ≥ · · · ≥ 2 ( a := n − p1 + 1 )

For p = p2 Nb ≥ Nb−1 + N0 ≥ Nb−2 + N1 ≥ · · · ≥ 2 ( b := n − p2 + 1 )

......

...

For p = pk−2 Nu ≥ Nu−1 + N0 ≥ Nu−2 + N1 ≥ · · · ≥ 2 ( u := n − pk−2 + 1 )

For p = pk−1 Nv ≥ Nv−1 + N0 ≥ Nv−2 + N1 ≥ · · · ≥ 2 ( v := n − pk−1 + 1 )

For p = pk Nw ≥ Nw−1 + N0 ≥ Nw−2 + N1 ≥ · · · ≥ 2 ( w := n − pk + 1 )

From the last row of inequalities it follows that Nw ≥ 2 . Since pk > pk−1 ,w :=n− pk + 1 and v := n− pk−1 + 1 it follows that v > w . Thus we may write theinequalities corresponding to the prime pk−1 as follows:

Nv ≥ Nv−1 +N0 ≥ Nv−2 +N1 ≥ · · · ≥ Nw +Nx ≥ · · · ≥ 2, (w ≥ x).

From this we get the result Nv ≥ 3 . (Because Nw ≥ 2 and Nx ≥ 1 ). Usingthe same argument, u > v , and the inequalities for the prime pk−2 (using thetheorem) could be written as

Nu ≥ Nu−1 +N0 ≥ Nn−2 +N1 · · · ≥ Nv +Ny ≥ · · · ≥ 2, (v ≥ y).

This yields the result Nu ≥ 4 . Moving up the rows in this manner finally givesthe result Na ≥ k + 1 . By the Wagner and Livingstone inequalities, Nm ≥ Na .�

Proposition 5.3.2 Let G be a group of degree n = p+ k ( k ∈ Z ) and let pbe the largest prime such that m < p ≤ n ( m := �n/2� ) If p does not dividethe order of G , then G cannot be (k + 1) -homogeneous.

Proof. The system of inequalities (5) and (6) both have the same first in-equalities, that is Nn−p+1 ≥ Nn−p + N0 and if we substitute n = p + k in thisinequality we get

N(p+k)−p+1 ≥ N(p+k)−p +N0 = Nk+1 ≥ Nk +N0 .

This means that Nk+1 ≥ 2 . Hence G is not (k + 1) -homogeneous. �The next result is of course extremely well-known. Still, I will show how it couldbe proved using the methods of modular homology.

72

Proposition 5.3.3 Let G be a permutation group of degree p , where p is aprime. Suppose that the order of G is not divisible by p . Then G is a non-transitive group.

Proof. Let us get the inequalities corresponding to the prime, p , which isequal to the order of G . We have the following

n = p, m = �p2� =

p− 1

2, r = n− p−m = p− p− (

p− 1

2) = −(

p− 1

2)

Thus, we have the following inequalities

N( p−12

)+N−( p−12

) ≥ N( p−12

)−1+N−( p−12

)+1 ≥ N( p−12

)−2+N−( p−12

)+2 ≥ · · · ≥ N1+N−1 ≥ 2N0

So N1 ≥ 2 . (Note that Nk = 0 , for k < 0 .) �

73

Chapter 6

Numbers of k -Orbits of Lineargroups

In this chapter I shall look at orbits under action of linear groups on the set ofk -dimensional subspaces of the vector space V (n, q) . It seems that no results,similar to the theorem of Polya, are known to enumerate orbits of linear groups.Some partial results belong to Zhe-Xian Wan [36, 37]. Nevertheless, methods ofmodular homology could be used to estimate numbers of such orbits.

The chapter is divided into two major sections. In the first section I shall lookat subgroups of the general linear groups. In the second section I will considerclassical subgroups of general linear groups.

6.1 Conditions for Nontriviality

In chapter 4, section (5.1), we looked at two results which hold for the numberNk(G) of k -orbits of permutation groups. I will state these results for k -orbitsof linear groups.

Proposition 6.1.1 Let G ≤ GLn(q) that acts on the k -dimensional subspacesof the vector space Vn(q) . Then the following inequalities hold:

1. [nk

]q

|G| ≤ Nk(G) ≤[nk

]q

.

Here[

nk

]q

is the number of k -dimensional subspaces of the vector space.

74

2. If[

nk

]q

does not divide |G| , then Nk(G) ≥ 2.

As in chapter 4, I will also state conditions for results to be non-trivial. Theresults will be called non-trivial if they do not follow directly from the two simpleresults stated above. Thus for a result, Nk ≥M , (where M is a positive integer)we must have these conditions in order for this result to be non-trivial.

1.

|G| >

[nk

]q

M,

2.[

nk

]q

must divide |G| in case M = 2.

We say that G has sufficiently large order if

|G| >

[nk

]q

M.

The theorem (4.10.9) stated in chapter 3 will play the key role in obtaining theresults in this chapter. I will restate this theorem here.

Theorem 6.1.2 Let V (n, q) be a vector space over GF(q) and let F be afield of prime characteristic p , not dividing q . Let Mk

n be the F -vector spacespanned by all the k -dimensional subspaces of V (n, q) . Also let G be a subgroupof GL(n, q) , whose order is co-prime to p . Then the following inequalities hold:

(Nm +Nr)π ≥ (Nm−1 +Nr+1)π ≥ · · · ≥ (Nm−s−1 +Nr+s+1)π.

where m = �n2� , r := n− p−m and s := �π

2�

6.2 Subgroups of the General Linear Groups

In this section I shall look at the k -orbits of some subgroups G of the generallinear group GL(n, q) . Namely, suppose that the order of GL(n, q) can bedecomposed in the product of primes as follows:

|GL(n, q)| = (qn − 1)(qn − q)(qn − q2)qn − q3) · · · (qn − qn−1) = pα11 p

α22 . . . pαs

s

Since order of G must divide |GL(n, q)| , only the same primes p1 , p2 ,. . . , ps

could appear in the decomposition of |G| .

75

I will consider subgroups G such that either one prime or two primes fromp1, . . . , ps are missed in the prime decomposition of the order of G . I will alsosuppose that such subgroups exist (though it could not be so in some cases).

To illustrate what sort of results I will get, let me begin by looking first at thegroup GL8(2) ,

|GL8(2)| = 228 · 35 · 52 · 72 · 17 · 31 · 127

= 5, 348, 063, 769, 211, 699, 200.

First, suppose that G is a subgroup of GL8(2) such that |G| is not divisibleby 5. Here n = 8 , π(5, 2) = 4 , m = �8

2� = 2 , r = n − π − m = 0 . So the

inequalities which we get are

N4 +N0 ≥ N3 +N1 ≥ 2N2.

If |G| is not divisible by 17, then r = 8 − 8 − 4 = −4 , (using π(17, 2) = 8 ).The inequalities that we obtain are

N4 +N−4 ≥ · · · ≥ N1 +N−1 ≥ 2N0.

This is equivalent to N1 ≥ 2 . This result is non-trivial if[

81

]2

= 255 divides

|G| . Thus we may conclude that

Proposition 6.2.1 Let G be a subgroup of the general linear group, GL8(2) .If |G| is not divisible by 17, then G is non-transitive on 1 -dimensional sub-spaces.

Next let us consider the case when G is a subgroup of GL8(2) , such that |G| isnot divisible by 127, π(127, 2) = 7 , r = 8− 7− 4 = −3 . The inequalities whichresult are

N4 +N−3 ≥ N3 +N−2 ≥ N2 +N−1 ≥ N1 +N0 ≥ 2

and this is equivalent to N2 ≥ 2. This result would be non-trivial if[

82

]2

=

10, 795 divides |G| .Now let us suppose that G is a of a subgroup of GL8(2) such that G is notdivisible by two primes 5 and 17. Then we will have two inequalities

N4 +N0 ≥ N3 +N1 ≥ 2N2

N1 ≥ 2.

Solving these inequalities gives the result N4 ≥ 3 . Thus, we have proved

76

Proposition 6.2.2 Every subgroup G of GL8(2) of order not divisible by5 × 17 = 85 has at least 3 orbits on 4 -dimensional subspaces of V8(2) .

The previous result will be non-trivial if |G| >[

84

]2

3= 66, 929 .

I have done more similar calculations. Results are summarized in the followingtables for subgroups of GLn(q) , for various values of n and q .

The results are general and hold for any subgroup of GLn(q) satisfying the con-ditions stated in the table. For any given pair of n and q , I have consideredtwo situations. Firstly, I have assumed that the subgroup G of GLn(q) doesnot have one prime in its prime decomposition as compared to the prime decom-position of GLn(q) . Then I have considered the situation when the subgroup Gdoes not have two primes in its prime decomposition.

I have also stated the values of π(p, q) at the beginning of the tables.

77

Group G < GL5(2) has one prime missed

|GL5(2)| = 210 · 32 · 5 · 7 · 31

= 9, 999, 360

Values of π(p, q) :

π(3, 2) = 2 , π(5, 2) = 4 , π(7, 2) = 3 , π(31, 2) = 5

|G| is not divisible by Results Condition for non-triviality

3 N2 +N1 ≥ 2N0

5 N2 ≥ 2 155||G|

7 N2 +N0 ≥ 2N1

31 N1 ≥ 2 31||G|

78

Group G < GL5(2) has two primes missed

|G| is not divisible by Results Condition for non-triviality

3 and 5 N2 +N1 ≥ 2N0

N2 ≥ 2 155||G|

3 and 7 N2 +N1 ≥ 2N0

N2 +N0 ≥ 2N1

3 and 31 N2 +N1 ≥ 2N0

N1 ≥ 2 31||G|

5 and 7 N2 +N0 ≥ 2N1

N2 ≥ 2 155||G|

5 and 31 N1 ≥ 2 31||G|

7 and 31 N2 ≥ 3 |G| > 51

79

Group G < GL5(3) has one prime missedGL5(3)

|GL5(3)| = 210 · 310 · 5 · 112 · 13

= 475, 566, 474, 240

Values of π(p, q) :

π(2, 3) = 2 , π(5, 3) = 4 , π(11, 3) = 5 , π(13, 3) = 3

|G| is not divisible by Results Condition for non-triviality

2 N2 +N1 ≥ 2N0

5 N2 ≥ 2 1, 210||G|

11 N1 ≥ 2 121||G|

31 N2 +N0 ≥ 2N1

80

Group G < GL5(3) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 5 N2 +N1 ≥ 2N0

N2 ≥ 2 1, 210||G|

2 and 11 N2 +N1 ≥ 2N0

N1 ≥ 2 121||G|

2 and 13 N2 +N1 ≥ 2N0

N2 +N0 ≥ 2N1

5 and 11 N1 ≥ 2 121||G|

5 and 13 N2 +N0 ≥ 2N1

N2 ≥ 2 1, 210||G|

11 and 13 N2 ≥ 3 |G| > 403

81

Group G < GL5(4) has one prime missed

|GL5(4)| = 220 · 36 · 52 · 7 · 11 · 17 · 31

= 775, 476, 766, 310, 400

Values of π(p, q) :

π(3, 4) = 3 , π(5, 4) = 2 , π(7, 4) = 3 , π(11, 4) = 5 , π(17, 4) = 4 ,π(31, 4) = 5

|G| is not divisible by Results Condition for non-triviality

3 N2 +N0 ≥ 2N1

5 N2 +N1 ≥ 2N0

7 N2 +N0 ≥ 2N1

11 N1 ≥ 2 341||G|

17 N2 ≥ 2 5, 797||G|

31 N1 ≥ 2 341||G|

82

Group G < GL5(4) has two primes missed

|G| is not divisible by Results Condition for non-triviality

3 and 5 N2 + N0 ≥ 2N1

N2 + N1 ≥ 2N0

3 and 7 N2 + N0 ≥ 2N1

3 and 11 N2 ≥ 3 |G| > 1, 932

3 and 17 N2 + N0 ≥ 2N1

N2 ≥ 2 5, 797||G|

3 and 31 N2 ≥ 3 |G| > 1, 932

5 and 7 N2 + N1 ≥ 2N0

N2 + N0 ≥ 2N1

5 and 11 N2 + N1 ≥ 2N0

N1 ≥ 2 341||G|

5 and 17 N2 + N1 ≥ 2N0

N2 ≥ 2 5, 797||G|

5 and 31 N2 + N1 ≥ 2N0

N1 ≥ 2 341||G|

7 and 11 N2 ≥ 3 |G| > 1, 932

7 and 17 N2 + N1 ≥ 2N0

N2 ≥ 2 5, 797||G|

7 and 31 N2 ≥ 3 |G| > 1, 932

11 and 17 N1 ≥ 2 341||G|

11 and 31 N1 ≥ 2 341||G|

17 and 31 N1 ≥ 2 341||G|

83

Group G < GL5(5) has one prime missed

|GL5(5)| = 213 · 32 · 510 · 11 · 13 · 31 · 71

= 226, 614, 960, 000, 000, 000

Values of π(p, q) :

π(2, 5) = 2 , π(3, 5) = 2 , π(11, 5) = 5 , π(13, 5) = 4 , π(31, 5) = 3 ,π(71, 5) = 5

|G| is not divisible by Results Condition for non-triviality

2 N2 +N1 ≥ 2N0

3 N2 +N1 ≥ 2N0

11 N1 ≥ 2 781||G|

13 N2 ≥ 2 20, 306||G|

31 N2 +N0 ≥ 2N1

71 N1 ≥ 2 781||G|

84

Group G < GL5(5) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 3 N2 + N1 ≥ 2N0

2 and 11 N2 + N1 ≥ 2N0

N1 ≥ 2 781||G|

2 and 13 N2 + N1 ≥ 2N0

N2 ≥ 2 20, 306||G|

2 and 31 N2 + N0 ≥ 2N1

N2 + N1 ≥ 2N0

2 and 71 N2 + N1 ≥ 2N0

N1 ≥ 2

3 and 11 N2 + N1 ≥ 2N0

N1 ≥ 2 781||G|

3 and 13 N2 + N1 ≥ 2N0

N2 ≥ 2 20, 306||G|

3 and 31 N2 + N1 ≥ 2N0

N2 + N0 ≥ 2N1

3 and 71 N2 + N1 ≥ 2N0

N1 ≥ 2 781||G|

11 and 13 N1 ≥ 2 781||G|

11 and 31 N2 ≥ 3 |G| > 6, 768

11 and 71 N1 ≥ 2 781||G|

13 and 31 N2 + N0 ≥ 2N1

N2 ≥ 2 20, 306||G|

13 and 71 N1 ≥ 2 781||G|

31 and 71 N2 ≥ 3 |G| > 6, 768

85

Group G < GL5(7) has one prime missed

|GL5(7)| = 212 · 36 · 52 · 710 · 19 · 2801

= 1, 122, 211, 189, 922, 928, 537, 600

Values of π(p, q) :

π(2, 7) = 2 , π(3, 7) = 3 , π(5, 7) = 4 , π(19, 7) = 3 , π(2801, 7) = 5

|G| is not divisible by Results Condition for non-triviality

2 N2 +N1 ≥ 2N0

3 N2 +N0 ≥ 2N1

5 N2 ≥ 2 140, 050||G|

19 N2 +N0 ≥ 2N1

2801 N1 ≥ 2 2, 801||G|

86

Group G < GL5(7) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 3 N2 +N1 ≥ 2N0

N2 +N0 ≥ 2N1

2 and 5 N2 +N1 ≥ 2N0

N2 ≥ 2 140, 050||G|

2 and 19 N2 +N1 ≥ 2N0

N2 +N0 ≥ 2N1

2 and 2801 N2 +N1 ≥ 2N0

N1 ≥ 2 2, 801||G|

3 and 5 N2 +N0 ≥ 2N1

N2 ≥ 2 140, 050||G|

3 and 19 N2 +N0 ≥ 2N0

3 and 2801 N2 ≥ 3 |G| > 46, 664

5 and 19 N2 +N0 ≥ 2N1

N2 ≥ 2 140, 050||G|

5 and 2801 N1 ≥ 2 2, 801||G|

19 and 2801 N2 ≥ 3 |G| > 46, 664

87

Group G < GL6(2) has one prime missed

|GL6(2)| = 215 · 34 · 5 · 72 · 31

= 20, 158, 709, 760

Values of π(p, q) :

π(3, 2) = 2 , π(5, 2) = 4 , π(7, 2) = 3 , π(31, 2) = 5

|G| is not divisible by Results Condition for non-triviality

3 N3 +N1 ≥ 2N2

5 N3 ≥ 2 1, 395||G|

7 N3 +N0 ≥ N2 +N1

31 N2 ≥ 2 651||G|

88

Group G < GL6(2) has two primes missed

|G| is not divisible by Results Condition for non-triviality

3 and 5 N3 +N1 ≥ 2N2

N3 ≥ 2 1, 395||G|

3 and 7 N3 +N1 ≥ 2N2

N3 +N0 ≥ N2 +N1

3 and 31 N3 ≥ 3 |G| > 465

5 and 7 N3 +N0 ≥ N2 +N1

N3 ≥ 2 1, 395||G|

5 and 31 N2 ≥ 2 651||G|

7 and 31 N3 +N0 ≥ N2 +N1

N2 ≥ 2 651||G|

89

Group G < GL6(3) has one prime missed

|GL6(3)| = 213 · 315 · 5 · 7 · 112 · 132

= 84, 129, 611, 558, 952, 960

Values of π(p, q) :

π(2, 3) = 2 , π(5, 3) = 4 , π(7, 3) = 6 , π(11, 3) = 5 , π(13, 3) = 3

|G| is not divisible by Results Condition for non-triviality

2 N3 +N1 ≥ 2N2

5 N3 ≥ 2 33, 880||G|

7 N1 ≥ 2 364||G|

11 N2 ≥ 2 11, 011||G|

13 N3 +N0 ≥ N2 +N1

90

Group G < GL6(3) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 5 N3 +N1 ≥ 2N2

N3 ≥ 2 33, 880||G|

2 and 7 N3 +N1 ≥ 2N2

N1 ≥ 2 364||G|

2 and 11 N3 +N1 ≥ 4

2 and 13 N3 +N1 ≥ 2N2

N3 +N0 ≥ N2 +N1

5 and 7 N1 ≥ 2 364||G|

5 and 11 N2 ≥ 2 11, 011||G|

5 and 13 N3 +N0 ≥ N2 +N1

N3 ≥ 2 33, 880||G|

7 and 11 N1 ≥ 2 364||G|

7 and 13 N3 ≥ 3 |G| > 11, 293

11 and 13 N3 ≥ 2 33, 880||G|

91

Group G < GL6(4) has one prime missed

|GL6(4)| = 230 · 38 · 53 · 72 · 11 · 13 · 17 · 31

= 3, 251, 791, 214, 634, 074, 112, 000

Values of π(p, q) :

π(3, 4) = 3 , π(5, 4) = 2 , π(7, 4) = 3 , π(11, 4) = 5 , π(13, 4) = 6 ,π(17, 4) = 4 , π(31, 4) = 5 ,

|G| is not divisible by Results Condition for non-triviality

3 N3 +N0 ≥ N2 +N1

5 N3 +N1 ≥ 2N2

7 N3 +N0 ≥ N2 +N1

11 N2 ≥ 2 93, 093|G|

13 N1 ≥ 2 1, 365||G|

17 N3 ≥ 2 376, 805||G|

31 N2 ≥ 2 93, 093|G|

92

Group G < GL6(4) has two primes missed

|G| is not divisible by Results Condition for non-triviality

3 and 5 N3 + N0 ≥ N2 + N1

N3 + N1 ≥ 2N2

3 and 7 N3 + N0 ≥ N2 + N1

3 and 11 N3 ≥ 2 376, 805||G|

3 and 13 N3 ≥ 3 |G| > 125, 601

3 and 17 N3 ≥ 2 376, 805||G|

3 and 31 N3 ≥ 2 376, 805||G|

5 and 13 N3 + N0 ≥ N2 + N1

N3 ≥ 2 376, 805||G|

5 and 7 N3 + N1 ≥ 2N2

N3 + N0 ≥ N2 + N1

5 and 11 N3 ≥ 3 |G| > 125, 601

5 and 13 N3 ≥ 2 376, 805||G|

5 and 17 N3 + N1 ≥ 2N2

N3 ≥ 2 376, 805||G|

5 and 31 N3 ≥ 3 |G| > 125, 601

7 and 11 N3 ≥ 2 376, 805||G|

7 and 13 N3 ≥ 3 |G| > 125, 601

7 and 17 N3 + N0 ≥ N2 + N1

N3 ≥ 2 376, 805||G|

7 and 31 N3 ≥ 2 376, 805||G|

11 and 13 N1 ≥ 2 1, 365||G|

11 and 17 N2 ≥ 2 93, 093||G|

11 and 31 N2 ≥ 2 93, 093||G|

13 and 17 N1 ≥ 2 1, 365||G|

13 and 31 N1 ≥ 2 1, 365||G|

17 and 31 N2 ≥ 2 93, 093||G|

93

Group G < GL6(5) has one prime missed

|GL6(5)| = 216 · 34 · 515 · 7 · 11 · 13 · 312 · 71

= 11, 064, 475, 422, 000, 000, 000, 000, 000

Values of π(p, q) :

π(2, 5) = 2 , π(3, 5) = 2 , π(7, 5) = 6 , π(11, 5) = 5 , π(13, 5) = 4 ,π(31, 5) = 3 , π(71, 5) = 5 ,

|G| is not divisible by Results Condition for non-triviality

2 N3 +N1 ≥ 2N2

3 N3 +N1 ≥ 2N2

7 N1 ≥ 2 3, 906||G|

11 N2 ≥ 2 508, 431||G|

13 N3 ≥ 2 2, 558, 556||G|

31 N3 +N0 ≥ N2 +N1

31 N2 ≥ 2 508, 431||G|

94

Group G < GL6(5) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 3 N3 + N1 ≥ 2N2

2 and 7 N3 ≥ 2 2, 558, 556||G|

2 and 11 N3 ≥ 3 |G| > 852, 852

2 and 13 N3 + N1 ≥ 2N2

N3 ≥ 2 2, 558, 556||G|

2 and 31 N3 + N1 ≥ 2N2

N3 + N0 ≥ N2 + N1

2 and 71 N3 ≥ 3 |G| > 852, 852

3 and 7 N3 ≥ 2 2, 558, 556||G|

3 and 11 N3 ≥ 3 |G| > 852, 852

3 and 13 N3 + N1 ≥ 2N2

N3 ≥ 2 2, 558, 556||G|

3 and 31 N3 + N1 ≥ 2N2

N3 + N0 ≥ N2 + N1

3 and 71 N3 ≥ 3 |G| > 852, 852

7 and 11 N1 ≥ 2 3, 906||G|

7 and 13 N1 ≥ 2 3, 906||G|

7 and 31 N3 ≥ 3 |G| > 852, 852

7 and 71 N1 ≥ 2 3, 906||G|

11 and 13 N2 ≥ 2 508, 431||G|

11 and 31 N3 ≥ 2 2, 558, 556||G|

11 and 71 N2 ≥ 2 508, 431||G|

13 and 31 N3 + N0 ≥ N2 + N1

N3 ≥ 2 2, 558, 556||G|

13 and 71 N2 ≥ 2 508, 431||G|

31 and 71 N3 ≥ 2 2, 558, 556||G|

95

Group G < GL6(7) has one prime missed

|GL6(7)| = 216 · 38 · 52 · 715 · 192 · 43 · 2801

= 2, 218, 959, 336, 124, 989, 671, 614, 429, 593, 600

Values of π(p, q) :

π(2, 7) = 2 , π(3, 7) = 3 , π(5, 7) = 4 , π(19, 7) = 3 , π(43, 7) = 6 ,π(2801, 7) = 5 .

|G| is not divisible by Results Condition for non-triviality

2 N3 +N1 ≥ 2N2

3 N3 +N0 ≥ N2 +N1

5 N3 ≥ 2 48, 177, 200||G|

19 N3 +N0 ≥ N2 +N1

43 N1 ≥ 2 19, 608||G|

2801 N2 ≥ 2 6, 865, 251||G|

96

Group G < GL6(7) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 3 N3 +N1 ≥ 2N2

N3 +N0 ≥ N2 +N1

2 and 5 N3 +N1 ≥ 2N2

N3 ≥ 2 48, 177, 200||G|

2 and 19 N3 +N1 ≥ 2N2

N3 +N0 ≥ N2 +N1

2 and 43 N3 ≥ 2 48, 177, 200||G|

2,2801 N3 ≥ 3 |G| > 16, 059, 066

3 and 5 N3 ≥ 2 48, 177, 200||G|

3 and 19 N3 +N0 ≥ N2 +N1

3 and 43 N3 ≥ 3 |G| > 16, 059, 066

3 and 2801 N3 ≥ 2 48, 177, 200||G|

5 and 19 N3 +N0 ≥ N2 +N1

N3 ≥ 2 48, 177, 200||G|

5 and 43 N1 ≥ 2 19, 608||G|

5 and 2801 N2 ≥ 2 6, 865, 251||G|

19 and 43 N3 ≥ 3 |G| > 16, 059, 066

19 and 2801 N3 ≥ 2 48, 177, 200||G|

43 and 2801 N1 ≥ 2 19, 608||G|

97

Group G < GL7(2) has one prime missed

|GL7(2)| = 221 · 34 · 5 · 72 · 31 · 127

= 163, 849, 992, 929, 280

Values of π(p, q) :

π(3, 2) = 2 , π(5, 2) = 4 , π(7, 2) = 3 , π(31, 2) = 5 , π(127, 2) = 7 .

|G| is not divisible by Results Condition for non-triviality

3 N3 +N2 ≥ 2

5 N3 +N0 ≥ N2 +N1

7 N3 +N1 ≥ 2N2

31 N3 ≥ 2 11811||G|

127 N1 ≥ 2 127||G|

98

Group G < GL7(2) has two primes missed

|G| is not divisible by Results Condition for non-triviality

3 and 5 N3 +N0 ≥ N2 +N1

3 and 7 N3 +N1 ≥ 2N2

3 and 31 N3 ≥ 2 11, 811||G|

3 and 127 N1 ≥ 2 127||G|

5 and 7 N3 +N0 ≥ N2 +N1

N3 +N1 ≥ 2N2

5 and 31 N3 +N0 ≥ N2 +N1

N3 ≥ 2 11, 811||G|

5 and 127 N3 ≥ 3 |G| > 3, 937

7 and 31 N3 +N1 ≥ 2N2

N3 ≥ 2 11, 811||G|

7 and 127 N3 ≥ 2 11, 811||G|

31 and 127 N1 ≥ 2 127||G|

99

Group G < GL7(3) has one prime missed

|GL7(3)| = 214 · 321 · 5 · 7 · 112 · 132 · 1093

= 134, 068, 444, 202, 678, 083, 338, 240

Values of π(p, q) :

π(3, 2) = 2 , π(5, 3) = 4 , π(7, 3) = 6 , π(11, 3) = 5 , π(13, 3) = 3 ,π(1093, 3) = 7

|G| is not divisible by Results Condition for non-triviality

2 N3 +N2 ≥ 2

5 N3 +N0 ≥ N2 +N1

7 N2 ≥ 2 99, 463||G|

11 N3 ≥ 2 925, 771||G|

13 N3 +N0 ≥ N2 +N1

1093 N1 ≥ 2 1, 093||G|

100

Group G < GL7(3) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 5 N3 +N0 ≥ N2 +N1

2 and 7 N2 ≥ 2 99, 463||G|

2 and 1 N3 ≥ 2 925, 771||G|

2 and 13 N3 +N0 ≥ N2 +N1

2 and 1093 N1 ≥ 2 1, 093||G|

5 and 7 N2 ≥ 2 99, 463||G|

5 and 11 N3 +N0 ≥ N2 +N1

N3 ≥ 2 925, 771||G|

5 and 13 N3 +N0 ≥ N2 +N1

5 and 1093 N3 ≥ 3 |G| > 308, 590

7 and 11 N2 ≥ 2 99, 463||G|

7 and 13 N2 ≥ 2 99, 463||G|

7and 1093 N2 ≥ 2 99, 463||G|

11 and 13 N3 +N0 ≥ N2 +N1

N3 ≥ 2 925, 771||G|

11 and 1093 N1 ≥ 2 1, 093||G|

13 and 1093 N3 ≥ 3 |G| > 308, 590

101

Group G < GL8(2) has one prime missed

|GL8(2)| = 228 · 35 · 52 · 72 · 17 · 31 · 127

= 5, 348, 063, 769, 211, 699, 200

Values of π(p, q) :

π(3, 2) = 2 , π(5, 2) = 4 , π(7, 2) = 3 , π(17, 2) = 8 , π(31, 2) = 5 ,π(127, 2) = 7

|G| is not divisible by Results Condition for non-triviality

3 N4 +N2 ≥ 2N3

5 N4 +N0 ≥ N3 +N1 ≥ 2N2

7 N4 +N1 ≥ N3 +N2

17 N1 ≥ 2 255||G|

31 N4 ≥ 2 200, 787||G|

127 N2 ≥ 2 10, 795||G|

102

Group G < GL8(2) has two primes missed

|G| is not divisible by Results Condition for non-triviality

3 and 5 N4 +N2 ≥ 2N3

N4 +N0 ≥ N3 +N1 ≥ 2N2

3 and 7 N4 +N2 ≥ 2N3

N4 +N1 ≥ N3 +N2

3 and 17 N2 ≥ 2 200, 787||G|

3 and 31 N4 +N2 ≥ 2N3

N4 ≥ 2 200787||G|

3 and 127 N4 ≥ 2 200, 787||G|

5 and 7 N4 +N0 ≥ N3 +N1 ≥ 2N2

N4 +N1 ≥ N3 +N2

5 and 17 N4 ≥ 3 |G| > 66, 929

5 and 31 N4 ≥ 2 200, 787||G|

5 and 127 N4 ≥ 3 |G| > 66, 929

7 and 17 N4 ≥ 2 200, 787||G|

7 and 31 N4 +N1 ≥ N3 +N2

N4 ≥ 2 200, 787||G|

7and 1 27 N4 ≥ 3 |G| > 66, 929

17 and 31 N1 ≥ 2 255||G|

17 and 127 N1 ≥ 2 255||G|

31 and 127 N2 ≥ 2 10, 795||G|

103

Group G < GL8(3) has one prime missed

|GL8(3)| = 219 · 328 · 52 · 7 · 112 · 132 · 41 · 1093

= 1, 923, 442, 429, 811, 445, 711, 790, 394, 572, 800

Values of π(p, q) :

π(2, 3) = 2 , π(5, 3) = 4 , π(7, 3) = 6 , π(11, 3) = 5 , π(13, 3) = 3 ,π(41, 3) = 8 , π(1093, 3) = 7 .

|G| is not divisible by Results Condition for non-triviality

2 N4 +N2 ≥ 2N3

5 N4 +N0 ≥ N3 +N1 ≥ 2N2

7 N3 ≥ 2 25, 095, 280||G|

11 N4 ≥ 2 75, 913, 222||G|

13 N4 +N1 ≥ N3 +N2

41 N1 ≥ 2 3, 280||G|

1093 N2 ≥ 2 896, 260||G|

104

Group G < GL8(3) has two primes missed

|G| is not divisible by Results Condition for non-triviality

2 and 5 N4 + N2 ≥ 2N3

N4 + N0 ≥ N3 + N1 ≥ 2N2

2 and 7 N4 ≥ 3 |G| > 25, 304, 407

2 and 11 N4 + N2 ≥ 2N3

N4 ≥ 2 75, 913, 222||G|

2 and 13 N4 + N2 ≥ 2N3

N4 + N1 ≥ N3 + N2

2 and 41 N4 ≥ 2 75, 913, 222||G|

2 and 1093 N4 ≥ 2 75, 913, 222||G|

5 and 7 N4 ≥ 2 75, 913, 222||G|

5 and 11 N4 + N0 ≥ N3 + N1 ≥ 2N2

N4 ≥ 2 75, 913, 222||G|

5 and 13 N4 + N0 ≥ N3 + N1 ≥ 2N2

N4 + N1 ≥ N3 + N2

5 and 41 N4 ≥ 3 |G| > 25, 304, 407

5 and 1093 N4 ≥ 3 |G| > 25, 304, 407

7 and 11 N3 ≥ 2 25, 095, 280||G|

7 and 13 N4 ≥ 2 75, 913, 222||G|

7 and 41 N1 ≥ 2 3, 280||G|

7 and 1093 N2 ≥ 2 896, 260||G|

11 and 13 N4 + N1 ≥ N3 + N2

N4 ≥ 2 75, 913, 222||G|

11 and 41 N1 ≥ 2 3, 280||G|

11 and 1093 N2 ≥ 2 896, 260||G|

13 and 41 N4 ≥ 2 75, 913, 222||G|

13 and 1093 N4 ≥ 3 |G| > 25, 304, 407

41 and 1093 N1 ≥ 2 3, 280||G|

6.3 Classical Linear Groups

The ATLAS [35] contains linear representations of classical groups. These linearrepresentations are of course subgroups of general linear groups. In this section I

105

shall use the technique of modular homology to obtain results about the numbersof orbits of such subgroups. It is important that now the existence of subgroupsis stated (in contrast with the previous Section).

I have found several classical groups for which the method of modular homologyworks and provides nontrivial results. I shall look at two such cases in full details,other results are in tables below.

Let us consider the alternating group A12 ,

|A12| = 29 · 35 · 52 · 7 · 11.

It is known from the Atlas that A12 has a representation of degree n = 10over the field GF (2) . Thus, m = �n

2� = 5 and q = 2 .

We will use the primes 17, 73 and 127 that do not divide |A12| . For these primeswe have π(17, 2) = 8 , π(73, 2) = 9 and π(127, 2) = 7 .

There are other primes which do not divide the order of A12 , however, the valueof π(p, q) in using these primes exceeds the degree of the representation and sothese primes would provide trivial results.

For p = 17 we have π(17, 2) = 8 and r = m− π = 5 − 8 = −3 . We obtain thefollowing inequalities.

(N5 +N−3)8 ≥ (N4 +N−2)8 ≥ (N3 +N−1)8 ≥ (N2 +N0)8 ≥ 2N1.

This implies that

N3 ≥ N2 +N0 ≥ 2N1. (6.1)

Next, using p = 73 we have π(73, 2) = 9 , r = m− π = 5 − 9 = −4 and obtainthe following inequalities:

(N5 +N−4)9 ≥ (N4 +N−3)9 ≥ (N3 +N−2)9 ≥ (N2 +N−1)9 ≥ (N1 +N0)9 ≥ 2

This implies thatN5 ≥ N4 ≥ N3 ≥ N2 ≥ N1 +N0 ≥ 2

orN2 ≥ 2. (6.2)

Finally, let p = 127 . Then we have π(127, 2) = 7 , r = m − π = 5 − 7 = −2and get the following:

(N5 +N−2)7 ≥ (N4 +N−1)7 ≥ (N3 +N0)7 ≥ (N2 +N1)7 ≥ (2N1)7.

From this we obtain

N4 ≥ N3 +N0 ≥ N2 +N1 ≥ 2N1. (6.3)

106

Thus we now have the following system of inequalities

N2 ≥ 2

N3 ≥ N2 +N0 ≥ 2N1

N4 ≥ N3 +N0 ≥ N2 +N1 ≥ 2N1.

Solving these inequalities gives the result N3 ≥ 3 and N4 ≥ 4 and both theseresults are non-trivial because A12 is of sufficiently large order. From these wecan conclude that

Proposition 6.3.1 A subgroup of GL10(2) , isomorphic to the alternatinggroup A12 , has at least 3 orbits on 3-subspaces and at least 4 orbits on 4-subspaces.

Next, consider the orthogonal group O−10(2) ,

|O−10(2)| = 220 · 36 · 52 · 7 · 11 · 17.

This group also has a representation of degree 10 over the field GF(2) . Weneed to find p such that the values of π(p, q) would satisfy the inequality 5 <π(p, q) ≤ 10 . For this we may use the primes 73 and 127. These two primes donot divide the order of O−

10(2) and π(73, 2) = 9 and also π(127, 2) = 7 .

Firstly using the prime 73 and π = 9 , we get r = m− π = 5− 9 = −4 Thus weget the following inequalities

N5 +N−4 ≥ N4 +N−3 ≥ N3 +N−2 ≥ N2 +N−1 ≥ N1 +N0 ≥ 2

This means thatN2 ≥ 2.

Next using the prime 127 and π = 7 , we get r = m − π = 5 − 7 = −2 we getand we also get the inequalities

N4 +N−1 ≥ N3 +N0 ≥ N2 +N1 ≥ 2N1. (6.4)

Solving the two inequalities we get the result N4 ≥ 3. This result is non-trivialbecause the order of O−

10(2) is sufficiently large.

Proposition 6.3.2 Any subgroup of GL10(2) isomorphic to the orthogonalgroup O−

10(2) has at least 3 orbits on 4-subspaces.

The tables which follow contain similar results about other classical groups. Inparticular, the previous Proposition is in the last line of the table in pages 113–114. Here n is the degree of the representation of the group while q is thecharacteristic of the field. I have also stated the values of π(p, q) .

107

nq

|G|i

snot

Exam

ple

sπ(p,q

)R

esult

sco

ndit

ion

for

hol

ds

for

div

isib

leby

non

-tri

via

lity

59

11,13

and

41U

5(3

)π(1

1,9)

=5

N1≥

273

81/|G

||U

5(3

)|=

211·3

10·5

·7·6

1

π(1

3,9)

=3

N2≥

3|G

|>20

1,74

8U

5(3

)

π(4

1,9)

=4

N2≥

3|G

|>20

1,74

8U

5(3

)

54

7,17

and

31U

5(2

)π(3

1,4)

=5

N1≥

234

1/|G

||U

5(2

)|=

210·3

5·5

·11

π(1

7,4)

=4

N2≥

3|G

|>1,

933

U5(3

)

π(7,4

)=

3N

2≥

3|G

|>1,

933

U5(3

)

55

11,13

and

31A

6π(1

1,5)

=5

N1≥

278

1/|G

||A

6|=

23·3

2·5

π(1

3,5)

=4

N2≥

3|G

|>6,

769

π(3

1,5)

=3

N2≥

3|G

|>6,

769

108

109

nq

|G|i

snot

Exam

ple

sπ(p,q

)R

esult

sco

ndit

ion

for

hol

ds

for

div

isib

leby

non

-tri

via

lity

62

5an

d31

L2(8

)π(3

1,2)

=5

N2≥

265

1/|G

||L

2(8

)|=

23·3

2·7

U3(3

)π(5,2

)=

4N

3≥

3|G

|>46

5U

3(3

)an

dL

2(8

)|U

3(3

)|=

23·3

2·7

64

13,1

7an

d31

M22

π(1

3,4)

=6

N1≥

21,

365/|G

||M

22|=

27·3

2·5

·7·1

1

π(3

1,4)

=5

N2≥

3|G

|>31,0

31M

22

π(1

7,4)

=4

N3≥

4|G

|>94,2

02M

22

64

11an

d17

G2(4

)π(1

1,4)

=5

N2≥

293,0

93/|G

||G

2(4

)|=

212·3

3·5

2·7

·13

π(1

7,4)

=4

N3≥

3|G

|>12

5,60

2G

2(4

)

63

7,11

and

13A

6π(7,3

)=

6N

1≥

236

4/|G

||A

6|=

23·3

2·5

π(1

1,3)

=5

N2≥

3|G

|>3,

671

π(1

3,3)

=3

N3≥

3|G

|>11,2

94

110

nq

|G|i

snot

Exam

ple

sπ(p,q

)R

esult

sco

ndit

ion

for

hol

ds

for

div

isib

leby

non

-tri

via

lity

73

5,11

and

13U

3(3

)π(1

1,3)

=5

N3≥

292

5,77

1/|G

||U

3(3

)|=

25·3

3·7

π(5,3

)=

4N

3+N

0≥N

2+N

1

π(1

3,3)

=3

N0+N

1+N

3≥

2N2

77

5,19

,29

and

43L

2(8

)π(2

9,7)

=7

N1≥

213

7,25

7/|G

||U

3(3

)|=

23·3

2·7

π(4

3,7)

=6

N2≥

3|G

|>11

2,13

8,96

9

π(5,7

)=

4N

3+N

0≥N

2+N

1

π(1

9,7)

=3

N0+N

1+N

3≥

6

82

17,31

and

127

S6(2

)π(1

7,2)

=8

N1≥

225

5/|G

||S

6(2

)|=

29·3

4·5

·7π(1

27,2

)=

7N

2≥

3|G

|>3,

599

S6(2

)

π(3

1,2)

=5

N4≥

5|G

|>40,1

58S

6(2

)

111

nq

|G|i

snot

Exam

ple

sπ(p

,q)

Res

ult

sco

nditio

nfo

rhold

sfo

rdiv

isib

leby

non-t

rivia

lity

82

31

and

127

S8(2

)π(1

27,2

)=

7N

2≥

210,7

95/|G

||S

8(2

)|=

216·3

5·5

2·7

·17

O− 8

(2)

π(3

1,2

)=

5N

4≥

3|G

|>66,9

29

S8(2

)and

O− 8

(2)

|O− 8

(2)|

=212·3

4·5

·7·1

7

84

11,1

3,4

3and

257

O− 8

(2)

π(2

57,4

)=

8N

1≥

221,8

54/|G

||O

− 8(2

)|=

212·3

4·5

·7·1

7

π(4

3,4

)=

7N

2≥

3|G

|>7,9

53,0

37

O− 8

(2)

π(1

3,4

)=

6N

3≥

4|G

|>387,7

10,5

21

π(1

1,4

)=

5N

4≥

5|G

|>1,2

44,3

22,7

08

85

3,1

1,3

1and

313

Sz(8

)π(3

13,5

)=

8N

1≥

297,6

56/|G

||S

z(8

)|=

26·5

·7·1

3π(1

1,5

)=

5N

4≥

4|G

|>501,3

13,2

12,0

52

π(3

1,5

)=

3N

4+

2N

1≥

N3

+N

2+

N0

π(3

,5)

=2

N4

+2N

2+

2N

1≥

2N

3+

2N

1

112

113

nq

|G|i

snot

Exam

ple

sπ(p,q

)R

esult

sco

ndit

ion

for

hol

ds

for

div

isib

leby

non

-tri

via

lity

92

7,17

,31

,73

and

127

A5

π(7

3,2)

=9

N1≥

251

1/|G

||A

5|=

22·3

·5π(1

7,2)

=8

N2≥

3|G

|>14,4

79

π(1

27,2

)=

7N

3≥

4|G

|>19

7,00

9

π(3

1,2)

=5

N4≥

5|G

|>66

1,95

0

π(7,2

)=

3N

4+N

2+N

1≥

10

93

11an

d75

7O

9(3

)π(7

57,3

)=

9N

1≥

29,

841/|G

||A

5|=

214·3

16·5

2·7

·13·4

1

π(1

1,3)

=5

N4≥

3|G

|>2,

058,

022,

087

O9(3

)

94

7,11

,13

,17

,19

and

43A

6π(1

9,4)

=9

N1≥

287,3

81/|G

||A

6|=

23·3

2·5

π(4

3,4)

=7

N3≥

4|G

|>24,8

19,4

38,1

40

π(1

3,4)

=6

N4≥

5|G

|>31

8,85

6,78

1,60

0

π(1

1,4)

=5

N4≥

5|G

|>31

8,85

6,78

1,60

0

π(1

7,4)

=4

N4+N

1+N

0≥N

3+N

2

π(7,4

)=

3N

4+N

3+N

1≥

9

114

nq

|G|i

snot

Exam

ple

sπ(p,q

)R

esult

sco

ndit

ion

for

hol

ds

for

div

isib

leby

non

-tri

via

lity

102

17,3

1,73

and

127

A12

π(7

3,2)

=9

N2≥

217

4,25

1/|G

||A

12|=

29·3

5·5

2·7

·11

π(1

7,2)

=8

N3≥

3|G

|>2,

115,

905

A12

π(1

27,2

)=

7N

4≥

4|G

|>13,4

35,9

97A

12

π(3

1,2)

=5

N5+

2N0≥

2N2

102

73an

d12

7S

10(2

)π(7

3,2)

=9

N2≥

217

4,25

1/|G

||S

10(2

)|=

225·3

6·5

2·7

·11·1

7·3

1

π(1

27,2

)=

7N

4≥

3|G

|>17,9

14,6

63S

10(2

)

102

31,73

and

127

O− 10(2

)π(7

3,2)

=9

N2≥

217

4,25

1/|G

||O

− 10(2

)|=

220·3

6·5

2·7

·11·1

7

π(1

27,2

)=

7N

4≥

3|G

|>17,9

14,6

63O

− 10(2

)

π(3

1,2)

=5

N5+

2N0≥

2N2

115

116

nq

|G|i

snot

Exam

ple

sπ(p,q

)R

esult

sco

ndit

ion

for

hol

ds

for

div

isib

leby

non

-tri

via

lity

102

11,73

and

127

O+ 10(2

)π(1

1,2)

=10

N1≥

210

23/|G

||O

+ 10(2

)|=

220·3

5·5

2·7

·17·3

1

π(7

3,2)

=9

N2≥

3|G

|>58,0

84O

+ 10(2

)

π(1

27,2

)=

7N

4≥

5|G

|>10,7

48,7

98O

+ 10(2

)

103

11,13

,41

and

757

U5(3

)π(7

57,3

)=

9N

2≥

272,6

36,4

21/|G

||U

5(3

)|=

211·3

10·5

·7·6

1

π(4

1,3)

=8

N3≥

3|G

|>6,

108,

909,

253

U5(3

)

π(1

1,3)

=5

N5≥

4|G

|>37

6,61

8,04

1,98

2

π(1

3,3)

=3

N5+

2N2≥

8

112

11,17

,23

,24

·A8

π(2

3,2)

=11

N1≥

22,

047/|G

|31

,73

and

127

|24·A

8|=

210·3

2·5

·7π(1

1,2)

=10

N2≥

3|G

|>23

2,67

624

·A8

π(7

3,2)

=9

N3≥

4|G

|>12,7

38,9

22

π(1

7,2)

=8

N4≥

5|G

|>17

3,25

0,30

2

π(1

27,2

)=

7N

5≥

6|G

|>59

1,47

2,80

4

π(3

1,2)

=5

N5+N

1≥

7

117

nq

|G|i

snot

Exam

ple

sπ(p

,q)

Res

ult

sco

nditio

nfo

rhold

sfo

rdiv

isib

leby

non-t

rivia

lity

12

217,23,

A14

π(2

3,2

)=

11

N2≥

22,7

94,1

55/|G

|31,73

and

127

|A14|=

210·3

5·5

2·7

2·1

1·1

3π(7

3,2

)=

9N

4≥

3|G

|>4,6

36,9

93,3

60

A14

π(1

7,2

)=

8N

5≥

4|G

|>28,6

07,2

57,4

30

A14

π(1

27,2

)=

7N

6≥

5|G

|>46,1

34,8

78,6

40

π(3

1,2

)=

5N

6+

N1≥

N4

+N

3

12

323,41,61

and

73

Suz

π(7

3,3

)=

12

N1≥

2265720/|G

||S

uz|=

213·3

7·5

2·7

·11·1

3π(2

3,3

)=

11

N2≥

3|G

|>1,9

61,3

01,4

63

suz

G2(4

)|G

2(4

)|=

212·3

3·5

2·7

·13

π(6

1,3

)=

10

N3≥

4|G

|>3,3

40,6

99,8

69,4

30

π(4

1,3

)=

8N

5≥

5|G

|>17,8

23,5

89,0

77,9

17,1

68

12

211,17,23

G2(4

)π(2

3,2

)=

11

N2≥

22,7

94,1

55/|G

|31,73

and

127

|G2(4

)|=

212·3

3·5

2·7

·13

π(1

1,2

)=

10

N3≥

3|G

|>136,1

15,2

65

G2(4

)

π(7

3,2

)=

9N

4≥

4|G

|>3,4

77,7

45,0

21

G2(4

)

π(1

7,2

)=

8N

5≥

5|G

|>22,8

85,8

05,9

43

π(1

27,2

)=

7N

6≥

6|G

|>38,4

45,7

32,2

06

π(3

1,2

)=

5N

6+

N1≥

6

118

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121