Finite groupsFrom Wikipedia, the free encyclopedia

Chapter 1

2E6 (mathematics)

In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form ofE6, depending on a quadratic extension of elds KL. Unfortunately the notation for the group is not standardized,as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L)(thinking of the group as a subgroup of E(L) xed by an outer involution).Over nite elds these groups form one of the 18 innite families of nite simple groups, and were introducedindependently by Tits (1958) and Steinberg (1959).

1.1 Over nite eldsThe group 2E6(q2) has order q36 (q12 1) (q9 + 1) (q8 1) (q6 1) (q5 + 1) (q2 1) /(3,q + 1). This is similar tothe order q36 (q12 1) (q9 1) (q8 1) (q6 1) (q5 1) (q2 1) /(3,q 1) of E6(q).Its Schur multiplier has order (3, q + 1) except for 2E6(22), when it has order 12 and is a product of cyclic groupsof orders 2,2,3. One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and theexceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.The outer automorphism group has order (3, q + 1) f where q2 = pf .

1.2 Over the real numbersOver the real numbers, 2E6 is the quasisplit form of E6, and is one of the ve real forms of E6 classied by lieCartan. Its maximal compact subgroup is of type F4.

1.3 References Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley &Sons, ISBN 978-0-471-50683-6, MR 0407163

Steinberg, Robert (1959), Variations on a theme of Chevalley, Pacic Journal of Mathematics 9: 875891,doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191

Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335 Tits, Jacques (1958), Les formes relles des groupes de type E6, Sminaire Bourbaki; 10e anne: 1957/1958.Textes des confrences; Exposs 152 168; 2e d. corrige, Expos 162 15, Paris: Secrtariat math'ematique,MR 0106247

2

Chapter 2

3-step group

In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classicationof CN groups and in the FeitThompson theorem. The denition of a 3-step group in these two cases is slightlydierent.

2.1 CN groupsIn the theory of CN groups, a 3-step group (for some prime p) is a group such that:

G = Op,p,p(G) Op,p(G) is a Frobenius group with kernel Op(G) G/Op(G) is a Frobenius group with kernel Op,p(G)/Op(G)

Any 3-step group is a solvable CN-group, and conversely any solvable CN-group is either nilpotent, or a Frobeniusgroup, or a 3-step group.Example: the symmetric group S4 is a 3-step group for the prime p=2.

2.2 Odd order groupsFeit & Thompson (1963, p.780) dened a three-step group to be a group G satisfying the following conditions:

The derived group of G is a Hall subgroup with a cyclic complement Q. If H is the maximal normal nilpotent Hall subgroup of G, then GHCG(H)G and HCG is nilpotent and His noncyclic.

For qQ nontrivial, CG(q) is cyclic and non-trivial and independent of q.

2.3 References Feit, Walter; Thompson, John G. (1963), Solvability of groups of odd order, Pacic Journal of Mathematics13: 7751029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, MR 0166261

Feit, Walter; Thompson, John G.; Hall, Marshall, Jr. (1960), Finite groups in which the centralizer of anynon-identity element is nilpotent,Mathematische Zeitschrift 74: 117, doi:10.1007/BF01180468, ISSN 0025-5874, MR 0114856

Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 81b:20002

3

Chapter 3

3-transposition group

In mathematical group theory, a 3-transposition group is a group generated by a conjugacy class of involutions,called the 3-transpositions, such that the product of any two involutions from the conjugacy class has order at most3. They were rst studied by Bernd Fischer (1964, 1970, 1971) who discovered the three Fischer groups as examplesof 3-transposition groups.

3.1 HistoryFischer (1964) rst studied 3-transposition groups in the special case when the product of any two distinct transpo-sitions has order 3. He showed that a nite group with this property is solvable, and has a (nilpotent) 3-group ofindex 2. Manin (1986) used these groups to construct examples of non-abelian CH-quasigroups and to describe thestructure of commutative Moufang loops of exponent 3.

3.2 Fischers theoremSuppose that G is a group that is generated by a conjugacy class D of 3-transpositions and such that the 2 and 3 coresO2(G) and O3(G) are both contained in the center Z(G) of G. Then Fischer (1971) proved that up to isomorphismG/Z(G) is one of the following groups and D is the image of the given conjugacy class:

G/Z(G) is the trivial group. G/Z(G) is a symmetric group S for n5, and D is the class of transpositions. (If n=6 there is a second classof 3-transpositions).

G/Z(G) is a symplectic group Spn(2) with n3 over the eld of order 2, and D is the class of transvections.(When n=2 there is a second class of transpositions.)

G/Z(G) is a projective special unitary group PSUn(2) with n5, and D is the class of transvections G/Z(G) is an orthogonal group On(2) with =1 and n4, and D is the class of transvections G/Z(G) is an index 2 subgroup POn,+(3) of the projective orthogonal group POn(3) (with =1 and n5)generated by the class D of reections of norm +1 vectors.

G/Z(G) is one of the three Fischer groups Fi22, Fi23, Fi24. G/Z(G) is one of two groups of the form 8+(2).S3 and P8+(3).S3, where stands for the derived subgroupof the orthogonal group and S3 is the group of diagram automorphisms for the D4 Dynkin diagram.

The missing cases with n small above either do not satisfy the condition about 2 and 3 cores or have exceptionalisomorphisms to other groups on the list.

4

3.3. PROPERTIES 5

3.3 PropertiesThe group Sn has order n! and for n>1 has a subgroup An of index 2 that is simple if n>4.The symmetric group Sn is a 3-transposition group for all n>1. The 3-transpositions are the elements that exchangetwo points, and leaving each of the remaining points xed. These elements are the transpositions (in the usual sense)of S. (For n=6 there is a second class of 3-transpositions, namely the class of the elements of S6 which are productsof 3 disjoint transpositions.)The symplectic group Spn(2) has order

2n2

(22 1)(24 1) (22n 1)

It is a 3-transposition group for all n1. It is simple if n>2, while for n=1 it is S3, and for n=2 it is S6 with a simplesubgroup of index 2, namely A6. The 3-transpositions are of the form xx+(x,v)v for non-zero v.The special unitary group SUn(2) has order

2n(n1)/2(22 1)(23 + 1) (2n (1)n)

The projective special unitary group PSUn(2) is the quotient of the special unitary group SUn(2) by the subgroupM of all the scalar linear transformations in SUn(2). The subgroup M is the center of SUn(2). Also, M has ordergcd(3,n).The group PSUn(2) is simple if n>3, while for n=2 it is S3 and for n=3 it has the structure 32:Q8 (Q8 = quaterniongroup).Both SUn(2) and PSUn(2) are 3-transposition groups for n=2 and for all n4. The 3-transpositions of SUn(2) forn=2 or n4 are of the form xx+(x,v)v for non-zero vectors v of zero norm. The 3-transpositions of PSUn(2)for n=2 or n4 are the images of the 3-transpositions of SUn(2) under the natural quotient map from SUn(2) toPSUn(2)=SUn(2)/M.The orthogonal group On(2) has order

2 2n(n1)(22 1)(24 1) (22n2 1)(2n 1)

(Over elds of characteristic 2, orthogonal group in odd dimensions are isomorphic to symplectic groups.) It has anindex 2 subgroup (sometimes denoted by n(2)), which is simple if n>2.The group On(2) is a 3-transposition group for all n>2 and =1. The 3-transpositions are of the form xx+(x,v)vfor vectors v such that Q(v)=1, where Q is the underlying quadratic form for the orthogonal group.The orthogonal groups On(3) are the automorphism groups of quadratic forms Q over the eld of 3 elements suchthat the discriminant of the bilinear form (a,,b)=Q(a+b)Q(a)Q(b) is 1. The group On,(3), where and aresigns, is the subgroup of On(3) generated by reections with respect to vectors v with Q(v)=+1 if is +, and is thesubgroup of On(3) generated by reections with respect to vectors v with Q(v)=1 if is .For =1 and =1, let POn,(3)=On,(3)/Z, where Z is the group of all scalar linear transformations in On,(3).If n>3, then Z is the center of On,(3).For =1, let n(3) be the derived subgroup of On(3). Let Pn(3)= n(3)/X, where X is the group of all scalarlinear transformations in n(3). If n>2, then X is the center of n(3).If n=2m+1 is odd the two orthogonal groups On(3) are isomorphic and have order

2 3m2(32 1)(34 1) (32m 1)

and On+,+(3) On,(3) (center order 1 for n>3), and On,+(3) On+,(3) (center order 2 for n>3), because the twoquadratic forms are scalar multiples of each other, up to linear equivalence.If n=2m is even the two orthogonal groups On(3) have orders

6 CHAPTER 3. 3-TRANSPOSITION GROUP

2 3m(m1)(32 1)(34 1) (32m2 1)(3m 1)and On+,+(3) On+,(3), and On,+(3) On,(3), because the two classes of transpositions are exchanged by anelement of the general orthogonal group that multiplies the quadratic form by a scalar. If n=2m, m>1 and m is even,then the centre of On+,+(3) On+,(3) has order 2, and the centre of On,+(3) On,(3) has order 1. If n=2m, m>2and m is odd, then the centre of On+,+(3) On+,(3) has order 1, and the centre of On,+(3) On,(3) has order 2.If n>3, and =1 and =1, the group On,(3) is a 3-transposition group. The 3-transpositions of the group On,(3)are of the form xx(x,v)v/Q(v)=x+(x, v)/(v,v) for vectors v with Q(v)=, where Q is the underlying quadratic formof On(3).If n>4, and =1 and =1, then On,(3) has index 2 in the orthogonal group On(3). The group On,(3) has asubgroup of index 2, namely n(3), which is simple modulo their centers (which have orders 1 or 2). In other words,Pn(3) is simple.If n>4 is odd, and (,)=(+,+) or (,), then On,+(3) and POn,+(3) are both isomorphic to SOn(3)=n(3):2,where SOn(3) is the special orthogonal group of the underlying quadratic form Q. Also, n(3) is isomorphic toPn(3), and is also non-abelian and simple.If n>4 is odd, and (,)=(+,) or (,+), then On,+(3) is isomorphic to n(3)2, and On,+(3) is isomorphic ton(3). Also, n(3) is isomorphic to Pn(3), and is also non-abelian and simple.If n>5 is even, and =1 and =1, then On,+(3) has the form n(3):2, and POn,+(3) has the form Pn(3):2.Also, Pn(3) is non-abelian and simple.Fi22 has order 217.39.52.7.11.13 = 64561751654400 and is simple.Fi23 has order 218.313.52.7.11.13.17.23 = 4089470473293004800 and is simple.Fi24 has order 222.316.52.73.11.13.17.23.29 and has a simple subgroup of index 2, namely Fi24'.

3.4 Isomorphisms and solvable casesThere are numerous degenerate (solvable) cases and isomorphisms between 3-transposition groups of small degreeas follows (Aschbacher 1997, p.46):

3.4.1 Solvable groupsThe following groups do not appear in the conclusion of Fishers theorem as they are solvable (with order a power of2 times a power of 3).

S1 = SU1(2) = PSU1(2) = O;1 (3) = PO

;1 (3) = PO

;1 (3) has order 1.

S2 = O+2 (2) = O

;1 (3) = O

+;2 (3) = PO

+;2 (3) = PO

;2 (3) has order 2, and it is a 3-

transposition group.O;2 (3) = PO

;3 (3) = 2

2 is elementary abelian of order 4, and it is not a 3-transposition group.S3 = Sp2(2) = SU2(2) = PSU2(2) = O

2 (2) has order 6, and it is a 3-transposition group.

O;3 (3) = 23 is elementary abelian of order 8, and it is not a 3-transposition group.

S4 = O;3 (3) = PO

;3 (3) has order 24, and it is a 3-transposition group.

PSU3(2) = 32 : Q8 has order 72, and it is not a 3-transposition group, where Q8 denotes the quaternion

group.O+4 (2) = (S3 S3) : 2 has order 72, and it is not a 3-transposition group.SU3(2) = 3

1+2 : Q8 has order 216, and it is not a 3-transposition group, where 31+2 denotes theextraspecial group of order 27 and exponent 3, and Q8 denotes the quaternion group.PO+;4 (3) = (A4 A4) : 2 has order 288, and it is not a 3-transposition group.O+;4 (3) = (SL2(3) SL2(3)) : 2 has order 576, where * denotes the central product, and it is not a3-transposition group.

3.5. PROOF 7

3.4.2 IsomorphismsThere are several further isomorphisms involving groups in the conclusion of Fischers theorem as follows. This listalso identies the Weyl groups of ADE Dynkin diagrams, which are all 3-transposition groups except W(D2)=22,with groups on Fischers list (W stands for Weyl group).

S5 = O4 (2) has order 120, and the group is a 3-transposition group.

S6 = Sp4(2) = O;4 (3) = PO

;4 (3) has order 720 (and 2 classes of 3-transpositions), and the

group is a 3-transposition group.S8 = O

+6 (2) = has order 40320, and the group is a 3-transposition group.

W (E6) = O6 (2) = O

;5 (3) = PO

;5 (3) has order 51840, and the group is a 3-transposition group.

PO;5 (3) = SU4(2) = PSU4(2) has order 25920, and the group is a 3-transposition group.W (E7) = 2 Sp6(2) has order 2903040, and the group is a 3-transposition group.W (E8) = 2:O

+8 (2) has order 69672960, and the group is a 3-transposition group.

O+;+4s (3) = O+;4s (3) = 2:PO

+;+4s (3) = 2:PO

+;4s (3) for all s1, and the group is a 3-transposition

group if s2.O;+4s (3) = O

;4s (3) = PO

;+4s (3) = PO

;4s (3) for all s1, and the group is a 3-transposition group

for all s1.O+;+4s+2(3) = O

+;4s+2(3) = 2:PO

+;+4s+2(3) = 2:PO

+;4s+2(3) for all s0, and the group is a 3-transposition

group for all s0.O;+4s+2(3) = O

;4s+2(3) = PO

;+4s+2(3) = PO

;4s+2(3) for all s0, and the group is a 3-transposition

group if s1.O+;+2m+1(3) = O

;2m+1(3) = PO

+;+2m+1(3) = PO

;2m+1(3) for allm0, and the group is a 3-transposition

group if m1.O;+2m+1(3) = O

+;2m+1(3) = 2 PO;+2m+1(3) = 2 PO+;2m+1(3) for all m0, and the group is a

3-transposition group if m=0 or m2.W (An) = Sn+1 for all n1, and the group is a 3-transposition group for all n1.W (Dn) = 2

n1:Sn for all n2, and the group is a 3-transposition group if n3.

3.5 ProofThe idea of the proof is as follows. Suppose that D is the class of 3-transpositions in G, and dD, and let H be thesubgroup generated by the set Dd of elements of D commuting with d. Then Dd is a set of 3-transpositions of H,so the 3-transposition groups can be classied by induction on the order by nding all possibilities for G given any3-transposition group H. For simplicity assume that the derived group of G is perfect (this condition is satised by allbut the two groups involving triality automorphisms.)

If O3(H) is not contained in Z(H) then G is the symmetric group S5 If O2(H) is not contained in Z(H) then L=H/O2(H) is a 3-transposition group, and L/Z(L) is either of typeSp(2n, 2) in which case G/Z(G) is of type Spn(2), or of type PSUn(2) in which case G/Z(G) is of typePSUn(2)

If H/Z(H) is of type Sn then either G is of type Sn or n = 6 and G is of type O6(2) If H/Z(H) is of type Spn(2) with 2n 6 then G is of type On(2) H/Z(H) cannot be of type On(2) for n 4. If H/Z(H) is of type POn, (3) for n>4 then G is of type POn, (3). If H/Z(H) is of type PSUn(2) for n 5 then n = 6 and G is of type Fi22 (and H is an exceptional double coverof PSU6(2))

8 CHAPTER 3. 3-TRANSPOSITION GROUP

If H/Z(H) is of type Fi22 then G is of type Fi23 and H is a double cover of Fi22. If H/Z(H) is of type Fi23 then G is of type Fi24 and H is the product of Fi23 and a group of order 2. H/Z(H) cannot be of type Fi24.

3.6 3-transpositions and graph theoryIt is fruitful to treat 3-transpositions as vertices of a graph. Join the pairs that do not commute, i. e. have a productof order 3. The graph is connected unless the group has a direct product decomposition. The graphs correspondingto the smallest symmetric groups are familiar graphs. The 3 transpositions of S3 form a triangle. The 6 transpositionsof S4 form an octahedron. The 10 transpositions of S5 form the complement of the Petersen graph.The symmetric group Sn can be generated by n1 transpositions: (1 2), (2 3), ..., (n1 n) and the graph of thisgenerating set is a straight line. It embodies sucient relations to dene the group Sn.[1]

3.7 References[1] Dickson, L. E. (2003) [1900], Linear Groups: With an Exposition of the Galois Field Theory, p. 287, ISBN 978-0-486-

49548-4

Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics 124, Cambridge Uni-versity Press, ISBN 978-0-521-57196-8, MR 1423599 contains a complete proof of Fischers theorem.

Fischer, Bernd (1964), Distributive Quasigruppen endlicher Ordnung, Mathematische Zeitschrift 83: 267303, doi:10.1007/BF01111162, ISSN 0025-5874, MR 0160845

Fischer, Bernd (1970), Finite groups generated by 3-transpositions, preprint, Coventry: Mathematics Institute,University of Warwick The rst part of this preprint (4 of 19 sections) was published as Fischer, Bernd (1971),Finite groups generated by 3-transpositions. I, InventionesMathematicae 13 (3): 232246, doi:10.1007/BF01404633,MR 0294487 The later part with the construction of the Fischer groups is still unpublished (as of 2014).

Manin, Yuri Ivanovich (1986) [1972], Cubic forms, North-Holland Mathematical Library 4 (2nd ed.), Ams-terdam: North-Holland, ISBN 978-0-444-87823-6, MR 833513

Weiss, Richard (1983), On Fischers characterization of Sp(2) and U(2)", Comm. Algebra 11 (22): 252754, doi:10.1080/00927878308822979, MR 0733341

Weiss, Richard (1985), A uniqueness lemma for groups generated by 3-transpositions, Math. Proc. Cam-bridge Philos. Soc. 97 (3): 421431, doi:10.1017/S030500410006299X, MR 0778676

Chapter 4

3D4

Inmathematics, the Steinberg triality groups of type 3D4 form a family of Steinberg or twisted Chevalley groups. Theyare quasi-split forms of D4, depending on a cubic Galois extension of eldsK L, and using the triality automorphismof the Dynkin diagram D4. Unfortunately the notation for the group is not standardized, as some authors write it as3D4(K) (thinking of 3D4 as an algebraic group taking values in K) and some as 3D4(L) (thinking of the group as asubgroup of D4(L) xed by an outer automorphism of order 3). The group 3D4 is very similar to an orthogonal orspin group in dimension 8.Over nite elds these groups form one of the 18 innite families of nite simple groups, and were introduced bySteinberg (1959).

4.1 ConstructionThe simply connected split algebraic group of type D4 has a triality automorphism of order 3 coming from an order3 automorphism of its Dynkin diagram. If L is a eld with an automorphism of order 3, then this induced an order3 automorphism of the group D4(L). The group 3D4(L) is the subgroup of D4(L) of points xed by . It has three8-dimensional representations over the eld L, permuted by the outer automorphism of order 3.

4.2 Over nite eldsThe group 3D4(q3) has order q12 (q8 + q4 + 1) (q6 1) (q2 1). For comparison, the split spin group D4(q) indimension 8 has order q12 (q8 2q4 + 1) (q6 1) (q2 1) and the quasisplit spin group 2D4(q2) in dimension 8 hasorder q12 (q8 1) (q6 1) (q2 1).The group 3D4(q3) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclicof order f where q3 = pf and p is prime.This group is also sometimes called 3D4(q), D42(q3), or a twisted Chevalley group.

4.3 3D4(23)The smallest member of this family of groups has several exceptional properties not shared by other members of thefamily. It has order 211341312 = 212347213 and outer automorphism group of order 3.The automorphism group of 3D4(23) is a maximal subgroup of the Thompson sporadic group, and is also a subgroupof the compact Lie group of type F4 of dimension 52. In particular it acts on the 26-dimensional representation of F4.In this representation it xes a 26-dimensional lattice that is the unique 26-dimensional even lattice of determinant 3with no norm 2 vectors, studied by Gross & Elkies (1996). The dual of this lattice has 819 pairs of vectors of norm8/3, on which 3D4(23) acts as a rank 4 permutation group.The group 3D4(23) has 9 classes of maximal subgroups, of structure

9

10 CHAPTER 4. 3D4

21+8:L2(8) xing a point of the rank 4 permutation representation on 819 points.[211]:(7 S3)U3(3):2S3 L2(8)(7 L2(7)):231+2.2S472:2A432:2A413:4

4.4 See also List of nite simple groups 2E6

4.5 References Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley &Sons, ISBN 978-0-471-50683-6, MR 0407163

Elkies, NoamD.; Gross, Benedict H. (1996), The exceptional cone and the Leech lattice, International Math-ematics Research Notices (14): 665698, doi:10.1155/S1073792896000426, ISSN 1073-7928, MR 1411589

Steinberg, Robert (1959), Variations on a theme of Chevalley, Pacic Journal of Mathematics 9: 875891,doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191

Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335

4.6 External links D(2) at the atlas of nite groups D(3) at the atlas of nite groups

Chapter 5

A-group

This article is about a type of mathematical group. For the third millennium BCNubian culture, see Nubian A-Group.

In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similarto abelian groups. The groups were rst studied in the 1940s by Philip Hall, and are still studied today. A great dealis known about their structure.

5.1 DenitionAn A-group is a nite group with the property that all of its Sylow subgroups are abelian.

5.2 HistoryThe term A-group was probably rst used in (Hall 1940, Sec. 9), where attention was restricted to soluble A-groups.Halls presentation was rather brief without proofs, but his remarks were soon expanded with proofs in (Taunt 1949).The representation theory of A-groups was studied in (It 1952). Carter then published an important relationshipbetween Carter subgroups and Halls work in (Carter 1962). The work of Hall, Taunt, and Carter was presented intextbook form in (Huppert 1967). The focus on soluble A-groups broadened, with the classication of nite simpleA-groups in (Walter 1969) which allowed generalizing Taunts work to nite groups in (Broshi 1971). Interest in A-groups also broadened due to an important relationship to varieties of groups discussed in (Ol'anski 1969). Moderninterest in A-groups was renewed when new enumeration techniques enabled tight asymptotic bounds on the numberof distinct isomorphism classes of A-groups in (Venkataraman 1997).

5.3 PropertiesThe following can be said about A-groups:

Every subgroup, quotient group, and direct product of A-groups are A-groups. Every nite abelian group is an A-group. A nite nilpotent group is an A-group if and only if it is abelian. The symmetric group on three points is an A-group that is not abelian. Every group of square-free order is an A-group. The derived length of an A-group can be arbitrarily large, but no larger than the number of distinct primedivisors of the order, stated in (Hall 1940), and presented in textbook form as (Huppert 1967, Kap. VI, Satz14.16).

11

polseGul seddel

12 CHAPTER 5. A-GROUP

The lower nilpotent series coincides with the derived series (Hall 1940). A soluble A-group has a unique maximal abelian normal subgroup (Hall 1940). The Fitting subgroup of a solvable A-group is equal to the direct product of the centers of the terms of thederived series, rst stated in (Hall 1940), then proven in (Taunt 1949), and presented in textbook form in(Huppert 1967, Kap. VI, Satz 14.8).

A non-abelian nite simple group is an A-group if and only if it is isomorphic to the rst Janko group or toPSL(2,q) where q > 3 and either q = 2n or q 3,5 mod 8, as shown in (Walter 1969).

All the groups in the variety generated by a nite group are nitely approximable if and only if that group isan A-group, as shown in (Ol'anski 1969).

Like Z-groups, whose Sylow subgroups are cyclic, A-groups can be easier to study than general nite groupsbecause of the restrictions on the local structure. For instance, a more precise enumeration of soluble A-groupswas found after an enumeration of soluble groups with xed, but arbitrary Sylow subgroups (Venkataraman1997). A more leisurely exposition is given in (Blackburn, Neumann & Venkataraman 2007, Ch. 12).

5.4 References Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of nite groups, Cam-bridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC154682311

Broshi, Aviad M. (1971), Finite groups whose Sylow subgroups are abelian, Journal of Algebra 17: 7482,doi:10.1016/0021-8693(71)90044-5, ISSN 0021-8693, MR 0269741

Carter, Roger W. (1962), Nilpotent self-normalizing subgroups and system normalizers, Proceedings of theLondon Mathematical Society. Third Series 12: 535563, doi:10.1112/plms/s3-12.1.535, MR 0140570

Hall, Philip (1940), The construction of soluble groups, Journal fr die reine und angewandte Mathematik182: 206214, ISSN 0075-4102, MR 0002877

Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050, especially Kap. VI, 14, p751760

It, Noboru (1952), Note on A-groups, Nagoya Mathematical Journal 4: 7981, ISSN 0027-7630, MR0047656

Ol'anski, A. Ju., Varieties of nitely approximable groups, Izvestiya Akademii Nauk SSSR. Seriya Matem-aticheskaya (in Russian) 33: 915927, ISSN 0373-2436, MR 0258927

Taunt, D. R. (1949), OnA-groups, Proc. Cambridge Philos. Soc. 45: 2442, doi:10.1017/S0305004100000414,MR 0027759

Venkataraman, Geetha (1997), Enumeration of nite soluble groups with abelian Sylow subgroups, TheQuarterly Journal ofMathematics. Second Series 48 (189): 107125, doi:10.1093/qmath/48.1.107,MR1439702

Walter, John H. (1969), The characterization of nite groups with abelian Sylow 2-subgroups., Annals ofMathematics. Second Series (TheAnnals ofMathematics, Vol. 89, No. 3) 89 (3): 405514, doi:10.2307/1970648,JSTOR 1970648, MR 0249504

Chapter 6

AlperinBrauerGorenstein theorem

In mathematics, theAlperinBrauerGorenstein theorem characterizes the nite simple groups with quasidihedralor wreathed[1] Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groupsor projective special unitary groups over a nite elds of odd order, depending on a certain congruence, or to theMathieu groupM11 . Alperin, Brauer & Gorenstein (1970) proved this in the course of 261 pages. The subdivisionby 2-fusion is sketched there, given as an exercise in Gorenstein (1968, Ch. 7), and presented in some detail in Kwonet al. (1980).

6.1 Notes[1] A 2-group is wreathed if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyclic

groups of the same order, that is, if it is the wreath product of a cyclic 2-group with the symmetric group on 2 points.

6.2 References Alperin, J. L.; Brauer, R.; Gorenstein, D. (1970), Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups., Transactions of the American Mathematical Society (American Mathematical Society) 151 (1):1261, doi:10.2307/1995627, ISSN 0002-9947, JSTOR 1995627, MR 0284499

Gorenstein, D. (1968), Finite groups, Harper & Row Publishers, MR 0231903 Kwon, T.; Lee, K.; Cho, I.; Park, S. (1980), On nite groups with quasidihedral Sylow 2-groups, Journal ofthe Korean Mathematical Society 17 (1): 9197, ISSN 0304-9914, MR 593804

13

Chapter 7

Alternating group

In mathematics, an alternating group is the group of even permutations of a nite set. The alternating group on theset {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by Anor Alt(n).

7.1 Basic properties

For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n !/ 2 elements. It is the kernel of the signature group homomorphism sgn : Sn {1, 1} explained under symmetricgroup.The group An is abelian if and only if n 3 and simple if and only if n = 3 or n 5. A5 is the smallest non-abeliansimple group, having order 60, and the smallest non-solvable group.The groupA4 has a Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions{(), (12)(34), (13)(24), (14)(23) }, and maps to A3 = C3, from the sequence V A4 A3 = C3. In Galois theory,this map, or rather the corresponding map S4 S3, corresponds to associating the Lagrange resolvent cubic to aquartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.

7.2 Conjugacy classes

As in the symmetric group, the conjugacy classes in An consist of elements with the same cycle shape. However, ifthe cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length oneare included in the cycle type, then there are exactly two conjugacy classes for this cycle shape (Scott 1987, 11.1,p299).Examples:

the two permutations (123) and (132) are not conjugates in A3, although they have the same cycle shape, andare therefore conjugate in S3

the permutation (123)(45678) is not conjugate to its inverse (132)(48765) inA8, although the two permutationshave the same cycle shape, so they are conjugate in S8.

7.3 Relation with symmetric group

see Symmetric group.

14

7.4. GENERATORS AND RELATIONS 15

7.4 Generators and relationsAn is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating setis often used to prove that An is simple for n 5 .

7.5 Automorphism groupFor more details on this topic, see Automorphisms of the symmetric and alternating groups.

For n > 3, except for n = 6, the automorphism group of An is the symmetric group Sn, with inner automorphism groupAn and outer automorphism group Z2; the outer automorphism comes from conjugation by an odd permutation.For n = 1 and 2, the automorphism group is trivial. For n = 3 the automorphism group is Z2, with trivial innerautomorphism group and outer automorphism group Z2.The outer automorphism group ofA6 is the Klein four-group V =Z2 Z2, and is related to the outer automorphism ofS6. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)).

7.6 Exceptional isomorphismsThere are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type,particularly projective special linear groups. These are:

A4 is isomorphic to PSL2(3)[1] and the symmetry group of chiral tetrahedral symmetry. A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry. (See[1] foran indirect isomorphism of PSL2(F5) A5 using a classication of simple groups of order 60, and here for adirect proof).

A6 is isomorphic to PSL2(9) and PSp4(2)' A8 is isomorphic to PSL4(2)

More obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group(which is also SL1(q) = PSL1(q) for any q).

7.7 Examples S4 and A4

7.8 SubgroupsA4 is the smallest group demonstrating that the converse of Lagranges theorem is not true in general: given a nitegroup G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d: the group G = A4, oforder 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects)with any additional element generates the whole group.

7.9 Group homologySee also: Symmetric group Homology

The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sucientlylarge n, it is constant. However, there are some low-dimensional exceptional homology. Note that the homology ofthe symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homologyelements).

16 CHAPTER 7. ALTERNATING GROUP

7.9.1 H1: Abelianization

The rst homology group coincides with abelianization, and (since An is perfect, except for the cited exceptions) isthus:

H1(An;Z) = 0 for n = 0; 1; 2 ;H1(A3;Z) = Aab3 = A3 = Z/3 ;H1(A4;Z) = Aab4 = Z/3 ;H1(An;Z) = 0 for n 5 .

This is easily seen directly, as follows. An is generated by 3-cycles so the only non-trivial abelianization maps areAn ! C3; since order 3 elements must map to order 3 elements and for n 5 all 3-cycles are conjugate, so theymust map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like(123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have orderdividing 2 and 3, so the abelianization is trivial.For n < 3 , An is trivial, and thus has trivial abelianization. For A3 and A4 one can compute the abelianizationdirectly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivialmaps A3 C3 (in fact an isomorphism) and A4 C3:

7.9.2 H2: Schur multipliers

Main article: Covering groups of the alternating and symmetric groups

The Schur multipliers of the alternating groups An (in the case where n is at least 5) are the cyclic groups of order 2,except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schurmultiplier is (the cyclic group) of order 6.[2] These were rst computed in (Schur 1911).

H2(An;Z) = 0 for n = 1; 2; 3 ;H2(An;Z) = Z/2 for n = 4; 5 ;H2(An;Z) = Z/6 for n = 6; 7 ;H2(An;Z) = Z/2 for n 8 .

7.10 Notes[1] Robinson (1996), p. 78

[2] Wilson, Robert (October 31, 2006), Chapter 2: Alternating groups, The nite simple groups, 2006 versions, 2.7: Coveringgroups

7.11 References

Robinson, Derek John Scott (1996), A course in the theory of groups, Graduate texts in mathematics 80 (2 ed.),Springer, ISBN 978-0-387-94461-6

Schur, Issai (1911), "ber die Darstellung der symmetrischen und der alternierendenGruppe durch gebrochenelineare Substitutionen, Journal fr die reine und angewandteMathematik 139: 155250, doi:10.1515/crll.1911.139.155

Scott, W.R. (1987), Group Theory, New York: Dover Publications, ISBN 978-0-486-65377-8

7.12. EXTERNAL LINKS 17

7.12 External links Weisstein, Eric W., Alternating group, MathWorld. Weisstein, Eric W., Alternating group graph, MathWorld.

Chapter 8

Aschbacher block

In mathematical nite group theory, a block, sometimes calledAschbacher block, is a subgroup giving an obstructionto Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher.

8.1 DenitionA group L is called short if it has the following properties (Aschbacher & Smith 2004, denition C.1.7):

1. L has no subgroup of index 2

2. The generalized Fitting subgroup F*(L) is a 2-group O2(L)3. The subgroup U = [O2(L), L] is an elementary abelian 2-group in the center of O2(L)

4. L/O2(L) is quasisimple or of order 35. L acts irreducibly on U/CU(L)

An example of a short group is the semidirect product of a quasisimple group with an irreducible module over the2-element eld F2A block of a group G is a short subnormal subgroup.

8.2 References Aschbacher, Michael (1981), Some results on pushing up in nite groups,Mathematische Zeitschrift 177 (1):6180, doi:10.1007/BF01214339, ISSN 0025-5874, MR 611470

Aschbacher, Michael; Smith, Stephen D. (2004), The classication of quasithin groups. I Structure of StronglyQuasithin K-groups, Mathematical Surveys and Monographs 111, Providence, R.I.: American MathematicalSociety, ISBN 978-0-8218-3410-7, MR 2097623

Foote, Richard (1980), Aschbacher blocks, The Santa Cruz Conference on Finite Groups (Univ. California,Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math. 37, Providence, R.I.: Amer. Math. Soc., pp. 3742,MR 604554

Solomon, Ronald (1980), Some results on standard blocks, The Santa Cruz Conference on Finite Groups(Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math. 37, Providence, R.I.: Amer. Math.Soc., MR 604555

18

Chapter 9

ATLAS of Finite Groups

The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway,Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computationalassistance from J. G. Thackray), published in December 1985 by Oxford University Press and reprinted with correc-tions in 2003 (ISBN 978-0-19-853199-9). It lists basic information about 93 nite simple groups, the informationbeing generally: its order, Schur multiplier, outer automorphism group, various constructions (such as presentations),conjugacy classes of maximal subgroups (with characters group action they dene), and, most importantly, charactertables (including power maps on the conjugacy classes) of the group itself and bicyclic extensions given by stemextensions and automorphism groups. In certain cases (such as for the Chevalley groups En(2) ), the character tableis not listed and only basic information is given.The ATLAS is a recognizable large format book (sized 420mm by 300mm) with a cherry red cardboard cover andspiral binding. The names of the authors, all six letters long, are printed on the cover in the form of an array whichevokes the idea of a character table.The ATLAS is being continued in the form of an electronic database, the ATLAS of Finite Group Representations.

19

Chapter 10

Automorphisms of the symmetric andalternating groups

In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groupsand alternating groups are both standard examples of these automorphisms, and objects of study in their own right,particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.

10.1 Summary

10.1.1 Generic case

n 6= 2; 6 : Aut(Sn) = Sn , and thus Out(Sn) = 1 .

Formally, Sn is complete and the natural map Sn ! Aut(Sn) is an isomorphism.

n 6= 2; 6 : Out(An) = Sn/An = C2 , and the outer automorphism is conjugation by an odd permutation.

n 6= 2; 3; 6 : Aut(An) = Aut(Sn) = Sn

Sn ! Aut(Sn)! Aut(An)

10.1.2 Exceptional cases

n = 1; 2 : trivial:

Aut(S1) = Out(S1) = Aut(A1) = Out(A1) = 1

Aut(S2) = Out(S2) = Aut(A2) = Out(A2) = 1

n = 3 : Aut(A3) = Out(A3) = S3/A3 = C2

n = 6 : Out(S6) = C2 , and Aut(S6) = S6 o C2 is a semidirect product.

n = 6 : Out(A6) = C2 C2 , and Aut(A6) = Aut(S6) = S6 o C2:

20

10.2. THE EXCEPTIONAL OUTER AUTOMORPHISM OF S6 21

10.2 The exceptional outer automorphism of S6Among symmetric groups, only S6 has a (non-trivial) outer automorphism, which one can call exceptional (in analogywith exceptional Lie algebras) or exotic. In fact, Out(S6) = C2.[2]

This was discovered by Otto Hlder in 1895.[3][4]

This also yields another outer automorphism of A6, and this is the only exceptional outer automorphism of a nitesimple group:[5] for the innite families of simple groups, there are formulas for the number of outer automorphisms,and the simple group of order 360, thought of as A6, would be expected to have two outer automorphisms, not four.However, when A6 is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadicgroups (not falling in an innite family), the notion of exceptional outer automorphism is ill-dened, as there is nogeneral formula.)

10.2.1 ConstructionThere are numerous constructions, listed in (Janusz & Rotman 1982).Note that as an outer automorphism, its a class of automorphisms, well-determined only up to an inner automorphism,hence there is not a natural one to write down.One method is:

Construct an exotic map (embedding) S5 S6 S6 acts by conjugation on the six conjugates of this subgroup;

yielding a map S6 SX, where X is the set of conjugates. Identifying X with the numbers 1, ..., 6(which depends on a choice of numbering of the conjugates, i.e., up to an element of S6 (an innerautomorphism)) yields an outer automorphism S6 S6.

This map is an outer automorphism, since a transposition doesn't map to a transposition, but inner automor-phisms preserve cycle structure.

Throughout the following, one can work with the multiplication action on cosets or the conjugation action on conju-gates.To see that S6 has an outer automorphism, recall that homomorphisms from a group G to a symmetric group Sn areessentially the same as actions of G on a set of n elements, and the subgroup xing a point is then a subgroup of indexat most n in G. Conversely if we have a subgroup of index n in G, the action on the cosets gives a transitive action ofG on n points, and therefore a homomorphism to Sn.

10.2.2 Exotic map S5 S6There is a subgroup (indeed, 6 conjugate subgroups) of S6 which are abstractly isomorphic to S5, and transitive assubgroups of S6.

Sylow 5-subgroups

Janusz and Rotman construct it thus:

S5 acts transitively by conjugation on its 6 Sylow 5-subgroups, yielding an embedding S5 S6 as a transitivesubgroup of order 120. (The obvious map Sn Sn xes a point and thus isn't transitive.)

This follows from inspection of 5-cycles: each 5-cycle generates a group of order 5 (thus a Sylow subgroup), there are5!/5 = 120/5 = 24 5-cycles, yielding 6 subgroups (as each subgroup also includes the identity), and Sn acts transitivelyby conjugation on cycles of a given class, hence transitively by conjugation on these subgroups.One can also use the Sylow theorems, which imply transitivity.

22 CHAPTER 10. AUTOMORPHISMS OF THE SYMMETRIC AND ALTERNATING GROUPS

PGL(2,5)

The projective linear group of dimension two over the nite eld with ve elements, PGL(2, 5), acts on the projectiveline over the eld with ve elements, P1(F5), which has six elements. Further, this action is faithful and 3-transitive,as is always the case for the action of the projective linear group on the projective line. This yields a map PGL(2, 5) S6 as a transitive subgroup. Identifying PGL(2, 5) with S5 and the projective special linear group PSL(2, 5) withA5 yields the desired exotic maps S5 S6 and A5 A6.[6]

Following the same philosophy, one can realize the outer automorphism as the following two inequivalent actions ofS6 on a set with six elements:[7]

the usual action as a permutation group;

the six inequivalent structures as the projective line P1(F5) the line has 6 points, and the projective lineargroup acts 3-transitively, so xing 3 of the points, there are 3! = 6 dierent ways to arrange the remaining 3points, which yields the desired alternative action.

Frobenius group

Another way: To construct an outer automorphism of S6, we need to construct an unusual subgroup of index 6 inS6, in other words one that is not one of the six obvious S5 subgroups xing a point (which just correspond to innerautomorphisms of S6).The Frobenius group of ane transformations of F5 (maps x 7! ax + bwhere a 0) has order 20 = (5 1) 5 and actson the eld with 5 elements, hence is a subgroup of S5. (Indeed, it is the normalizer of a Sylow 5-group mentionedabove, thought of as the order-5 group of translations of F5.)S5 acts transitively on the coset space, which is a set of 120/20 = 6 elements (or by conjugation, which yields theaction above).

10.2.3 Other constructions

Ernst Witt found a copy of Aut(S6) in the Mathieu group M12 (a subgroup T isomorphic to S6 and an element that normalizes T and acts by outer automorphism). Similarly to S6 acting on a set of 6 elements in 2 dierent ways(having an outer automorphism), M12 acts on a set of 12 elements in 2 dierent ways (has an outer automorphism),though since M12 is itself exceptional, one does not consider this outer automorphism to be exceptional itself.The full automorphism group of A6 appears naturally as a maximal subgroup of the Mathieu group M12 in 2 ways,as either a subgroup xing a division of the 12 points into a pair of 6-element sets, or as a subgroup xing a subset of2 points.Another way to see that S6 has a nontrivial outer automorphism is to use the fact that A6 is isomorphic to PSL2(9),whose automorphism group is the projective semilinear group PL2(9), in which PSL2(9) is of index 4, yielding anouter automorphism group of order 4. The most visual way to see this automorphism is to give an interpretation viaalgebraic geometry over nite elds, as follows. Consider the action of S6 on ane 6-space over the eld k with 3elements. This action preserves several things: the hyperplane H on which the coordinates sum to 0, the line L in Hwhere all coordinates coincide, and the quadratic form q given by the sum of the squares of all 6 coordinates. Therestriction of q to H has defect line L, so there is an induced quadratic form Q on the 4-dimensional H/L that onechecks is non-degenerate and non-split. The zero scheme of Q in H/L denes a smooth quadric surface X in theassociated projective 3-space over k. Over an algebraic closure of k, X is a product of two projective lines, so by adescent argument X is the Weil restriction to k of the projective line over a quadratic tale algebra K. Since Q is notsplit over k, an auxiliary argument with special orthogonal groups over k forces K to be a eld (rather than a productof two copies of k). The natural S6-action on everything in sight denes a map from S6 to the k-automorphism groupof X, which is the semi-direct product G of PGL2(K) = PGL2(9) against the Galois involution. This map carries thesimple group A6 nontrivially into (hence onto) the subgroup PSL2(9) of index 4 in the semi-direct product G, so S6is thereby identied as an index-2 subgroup of G (namely, the subgroup of G generated by PSL2(9) and the Galoisinvolution). Conjugation by any element of G outside of S6 denes the nontrivial outer automorphism of S6.

10.3. NO OTHER OUTER AUTOMORPHISMS 23

10.2.4 Structure of outer automorphism

On cycles, it exchanges permutations of type (12) with (12)(34)(56) (class 21 with class 23), and of type (123) with(145)(263) (class 31 with class 32). The outer automorphism also exchanges permutations of type (12)(345) with(123456) (class 2131 with class 61). For each of the other cycle types in S6, the outer automorphism xes the classof permutations of the cycle type.On A6, it interchanges the 3-cycles (like (123)) with elements of class 32 (like (123)(456)).

10.3 No other outer automorphismsTo see that none of the other symmetric groups have outer automorphisms, it is easiest to proceed in two steps:

1. First, show that any automorphism that preserves the conjugacy class of transpositions is an inner automor-phism. (This also shows that the outer automorphism of S6 is unique; see below.) Note that an automorphismmust send each conjugacy class (characterized by the cyclic structure that its elements share) to a (possiblydierent) conjugacy class.

2. Second, show that every automorphism (other than the above for S6) stabilizes the class of transpositions.

The latter can be shown in two ways:

For every symmetric group other than S6, there is no other conjugacy class of elements of order 2 with thesame number of elements as the class of transpositions.

Or as follows:

Each permutation of order two (called an involution) is a product of k>0 disjoint transpositions, so it has cyclicstructure 2k1n-2k. Whats special about the class of transpositions (k=1)?If one forms the product of two dierent transpositions 1 and 2, then one always obtains either a 3-cycle or apermutation of type 221n4, so the order of the produced element is either 2 or 3. On the other hand if one forms aproduct of two involutions 1, 2> that have type k>1, sometimes it happens that the product contains either

two 2-cycles and a 3-cycle (for k=2 and n 7)

a 7-cycle (for k=3 and n 7)

two 4-cycles (for k=4 and n 8)

(for larger k, add to the permutations 1, 2 of the last example redundant 2-cycles that cancel each other). Now onearrives at a contradiction, because if the class of transpositions is sent via the automorphism f to a class of involutionsthat has k>1, then there exist two transpositions 1, 2 such that f(12)=f(1)f(2) has order 6, 7 or 4, but we knowthat 12 has order 2 or 3.

10.3.1 No other outer automorphisms of S6S6 has exactly one (class) of outer automorphisms: Out(S6) = C2.To see this, observe that there are only two conjugacy classes of S6 of order 15: the transpositions and those ofclass 23. Thus Aut(S6) acts on these two conjugacy classes (and the outer automorphism above interchanges theseconjugacy classes), and an index 2 subgroup stabilizes the transpositions. But an automorphism that stabilizes thetranspositions is inner, so the inner automorphisms are an index 2 subgroup of Aut(S6), so Out(S6) = C2.More pithily: an automorphism that stabilizes transpositions is inner, and there are only two conjugacy classes oforder 15 (transpositions and triple transpositions), hence the outer automorphism group is at most order 2.

24 CHAPTER 10. AUTOMORPHISMS OF THE SYMMETRIC AND ALTERNATING GROUPS

10.4 Small n

10.4.1 SymmetricFor n = 2, S2 = C2 = Z/2 and the automorphism group is trivial (obviously, but more formally because Aut(Z/2) =GL(1, Z/2) = Z/2* = 1). The inner automorphism group is thus also trivial (also because S2 is abelian).

10.4.2 AlternatingFor n = 1 and 2, A1 = A2 = 1 is trivial, so the automorphism group is also trivial. For n = 3, A3 = C3 = Z/3 is abelian(and cyclic): the automorphism group is GL(1, Z/3*) = C2, and the inner automorphism group is trivial (because itis abelian).

10.5 Notes[1] Janusz, Gerald; Rotman, Joseph (JuneJuly 1982), Outer Automorphisms of S6", The American Mathematical Monthly

89 (6): 407410, JSTOR 2321657

[2] Lam, T. Y., &Leep, D. B. (1993). Combinatorial structure on the automorphism group of S6". ExpositionesMathematicae,11(4), 289308.

[3] Lam, T. Y., &Leep, D. B. (1993). Combinatorial structure on the automorphism group of S6". ExpositionesMathematicae,11(4), 289308.

[4] Otto Hlder (1895), Bildung zusammengesetzter Gruppen, Mathematische Annalen, 46, 321422.

[5] ATLAS p. xvi

[6] Carnahan, Scott (2007-10-27), Small nite sets, Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre.

[7] Snyder, Noah (2007-10-28), The Outer Automorphism of S6", Secret Blogging Seminar

10.6 References http://polyomino.f2s.com/david/haskell/outers6.html Some Thoughts on the Number 6, by John Baez: relates outer automorphism to icosahedron 12 points in PG(3, 5) with 95040 self-transformations in The Beauty of Geometry, by Coxeter: discussesouter automorphism on rst 2 pages

http://links.jstor.org/sici?sici=0002-9890(198206%2F07)89%3A6%3C407%3AOAO%3E2.0.CO%3B2-L http://links.jstor.org/sici?sici=0002-9890(199304)100%3A4%3C377%3ASOTCAO%3E2.0.CO%3B2-S http://links.jstor.org/sici?sici=0002-9890(196606%2F07)73%3A6%3C642%3ATOAO%3E2.0.CO%3B2-P http://links.jstor.org/sici?sici=0002-9890(195804)65%3A4%3C252%3AOATOH%3E2.0.CO%3B2-I

Chapter 11

B-theorem

In mathematical nite group theory, the B-theorem (formerly the B-conjecture) states that if C is the centralizer ofan involution of a nite group, then every component of C/O(C) is the image of a component of C (Gorenstein 1983,p. 7 and chapter 3).

11.1 References Gorenstein, D. (1983), The classication of nite simple groups. Vol. 1, The University Series in Mathematics,Plenum Press, ISBN 978-0-306-41305-6, MR 746470

25

Chapter 12

Baby Monster group

In the area of modern algebra known as group theory, the baby monster group B (or, more simply, the babymonster) is a sporadic simple group of order

241 313 56 72 11 13 17 19 23 31 47= 4154781481226426191177580544000000= 4,154,781,481,226,426,191,177,580,544,000,000 41033.

12.1 HistoryB is one of the 26 sporadic groups and has the second highest order of these, with the highest order being that ofthe monster group. The double cover of the baby monster is the centralizer of an element of order 2 in the monstergroup. The outer automorphism group is trivial and the Schur multiplier has order 2.The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during hisinvestigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product ofany two elements has order at most 4. He investigated its properties and computed its character table. The rstconstruction of the baby monster was later realized as a permutation group on 13 571 955 000 points using a computerby Jerey Leon and Charles Sims,[1][2] though Robert Griess later found a computer-free construction using the factthat its double cover is contained in the monster. The name baby monster was suggested by John Horton Conway.[3]

12.2 RepresentationsIn characteristic 0 the 4371-dimensional representation of the baby monster does not have a nontrivial invariantalgebra structure analogous to the Griess algebra, but Ryba (2007) showed that it does have such an invariant algebrastructure if it is reduced modulo 2.The smallest faithful matrix representation of the Baby Monster is of size 4370 over the nite eld of order 2.Hhn (1996) constructed a vertex operator algebra acted on by the baby monster.

12.3 Generalized Monstrous MoonshineConway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, butthat similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one canconstruct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For theBaby monster B or F2, the relevant McKay-Thompson series is T2A() where one can set the constant term a(0) =104 ( A007267),

26

12.4. MAXIMAL SUBGROUPS 27

j2A() = T2A() + 104

= ()

(2)

12+ 26

(2)()

122=

1

q+ 104 + 4372q + 96256q2 + 1240002q3 + 10698752q4 : : :

and () is the Dedekind eta function.

12.4 Maximal subgroupsWilson (1999) found the 30 conjugacy classes of maximal subgroups of B as follows:

2.2E6(2):2 This is the centralizer of an involution, and is the subgroup xing a point of the smallest permutationrepresentation on 13 571 955 000 points.

21+22.Co2 Fi23 29+16.S8(2) Th (22 F4(2)):2 22+10+20.(M22:2 S3) [230].L5(2) S3 Fi22:2 [235].(S5 L3(2)) HN:2 O8+(3):S4 31+8.21+6.U4(2).2 (32:D8 U4(3).2.2).2 5:4 HS:2 S4 2F4(2) [311].(S4 2S4) S5 M22:2 (S6 L3(4):2).2 53.L3(5) 51+4.21+4.A5.4 (S6 S6).4 52:4S4 S5 L2(49).23 L2(31) M11

28 CHAPTER 12. BABY MONSTER GROUP

L3(3) L2(17):2 L2(11):2 47:23

12.5 References[1] (Gorenstein 1993)

[2] Leon, Jerey S.; Sims, Charles C. (1977). The existence and uniqueness of a simple group generated by {3,4}-transpositions.Bull. Amer. Math. Soc. 83 (5): 10391040. doi:10.1090/s0002-9904-1977-14369-3.

[3] Ronan, Mark (2006). Symmetry and the monster. Oxford University Press. pp. 178179. ISBN 0-19-280722-6.

Gorenstein, D. (1993), A brief history of the sporadic simple groups, in Corwin, L.; Gelfand, I. M.; Lep-owsky, James, The Gelfand Mathematical Seminars, 19901992, Boston, MA: Birkhuser Boston, pp. 137143, ISBN 978-0-8176-3689-0, MR 1247286

Hhn, Gerald (1996), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner MathematischeSchriften [BonnMathematical Publications], 286, Bonn: Universitt BonnMathematisches Institut, arXiv:0706.0236,MR 1614941

Ryba, Alexander J. E. (2007), A natural invariant algebra for the Baby Monster group, Journal of GroupTheory 10 (1): 5569, doi:10.1515/JGT.2007.006, MR 2288459

Wilson, Robert A. (1999), The maximal subgroups of the Baby Monster. I, Journal of Algebra 211 (1):114, doi:10.1006/jabr.1998.7601, MR 1656568

12.6 External links MathWorld: Baby monster group Atlas of Finite Group Representations: Baby Monster group

Chapter 13

BaerSuzuki theorem

In mathematical nite group theory, the BaerSuzuki theorem, proved by Baer (1957) and Suzuki (1965), statesthat if any two elements of a conjugacy class C of a nite group generate a nilpotent subgroup, then all elements ofthe conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a short elementary proof.

13.1 References Alperin, J. L.; Lyons, Richard (1971), On conjugacy classes of p-elements, Journal of Algebra 19: 536537,doi:10.1016/0021-8693(71)90086-x, ISSN 0021-8693, MR 0289622

Baer, Reinhold (1957), Engelsche Elemente Noetherscher Gruppen,Mathematische Annalen 133: 256270,doi:10.1007/BF02547953, ISSN 0025-5831, MR 0086815

Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6,MR 569209

Suzuki, Michio (1965), Finite groups in which the centralizer of any element of order 2 is 2-closed, Annalsof Mathematics. Second Series 82: 191212, ISSN 0003-486X, JSTOR 1970569, MR 0183773

29

Chapter 14

Balance theorem

In mathematical nite group theory, the balance theorem states that if G is a group with no core then G either hasdisconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type (Gorenstein & 1983 p. 7).The signicance of this theorem is that it splits the classication of nite simple groups into three major subcases.

14.1 References Gorenstein, D. (1983), The classication of nite simple groups. Vol. 1, The University Series in Mathematics,Plenum Press, ISBN 978-0-306-41305-6, MR 746470

30

Chapter 15

Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, C2n , thought of as an extensionof the cyclic group Cn by a cyclic group of order 2. It is the binary polyhedral group corresponding to the cyclicgroup.[1]

In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations (Cn < SO(3) ) under the 2:1 covering homomorphism

Spin(3)! SO(3)

of the special orthogonal group by the spin group.As a subgroup of the spin group, the binary cyclic group can be described concretely as a discrete subgroup of the unitquaternions, under the isomorphism Spin(3) = Sp(1) where Sp(1) is the multiplicative group of unit quaternions.(For a description of this homomorphism see the article on quaternions and spatial rotations.)

15.1 See also binary dihedral group binary tetrahedral group binary octahedral group binary icosahedral group

15.2 References[1] Coxeter, H. S. M. (1959), Symmetrical denitions for the binary polyhedral groups, Proc. Sympos. Pure Math., Vol. 1,

Providence, R.I.: American Mathematical Society, pp. 6487, MR 0116055.

31

Chapter 16

Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n[1]) is a member of a class of non-abelian groups of order4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic.In the notation of exact sequences of groups, this extension can be expressed as:

1! C2n ! Dicn ! C2 ! 1:More generally, given any nite abelian group with an order-2 element, one can dene a dicyclic group.

16.1 DenitionFor each integer n > 1, the dicyclic group Dicn can be dened as the subgroup of the unit quaternions generated by

a = ei/n = cos n+ i sin

nx = j

More abstractly, one can dene the dicyclic group Dicn as any group having the presentation[2]

Dicn = ha; x j a2n = 1; x2 = an; x1ax = a1i:Some things to note which follow from this denition:

x4 = 1 x2ak = ak+n = akx2

if j = 1, then xjak = akxj . akx1 = aknanx1 = aknx2x1 = aknx.

Thus, every element of Dicn can be uniquely written as akxj , where 0 k < 2n and j = 0 or 1. The multiplicationrules are given by

akam = ak+m

akamx = ak+mx akxam = akmx akxamx = akm+n

32

16.2. PROPERTIES 33

It follows that Dicn has order 4n.[2]

When n = 2, the dicyclic group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, thedicyclic group is isomorphic to the generalized quaternion group.[2]

16.2 PropertiesFor each n > 1, the dicyclic group Dicn is a non-abelian group of order 4n. (Dic1" is C4, the cyclic group of order4, which is abelian, and is not considered dicyclic.)Let A = be the subgroup of Dicn generated by a. Then A is a cyclic group of order 2n, so [Dicn:A] = 2. As asubgroup of index 2 it is automatically a normal subgroup. The quotient group Dicn/A is a cyclic group of order 2.Dicn is solvable; note that A is normal, and being abelian, is itself solvable.

16.3 Binary dihedral group

The dicyclic group is a binary polyhedral group it is one of the classes of subgroups of the Pin group Pin(2),which is a subgroup of the Spin group Spin(3) and in this context is known as the binary dihedral group.The connection with the binary cyclic group Cn, the cyclic group Cn, and the dihedral group Dihn of order 2n isillustrated in the diagram at right, and parallels the corresponding diagram for the Pin group.There is a supercial resemblance between the dicyclic groups and dihedral groups; both are a sort of mirroring ofan underlying cyclic group. But the presentation of a dihedral group would have x2 = 1, instead of x2 = an; and thisyields a dierent structure. In particular, Dicn is not a semidirect product of A and , since A is not trivial.The dicyclic group has a unique involution (i.e. an element of order 2), namely x2 = an. Note that this element liesin the center of Dicn. Indeed, the center consists solely of the identity element and x2. If we add the relation x2 = 1to the presentation of Dicn one obtains a presentation of the dihedral group Dihn, so the quotient group Dicn/is isomorphic to Dihn.There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation groupdescribed at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions

34 CHAPTER 16. DICYCLIC GROUP

one can ask what the image of it is under this homomorphism. The answer is just the dihedral symmetry group Dihn.For this reason the dicyclic group is also known as the binary dihedral group. Note that the dicyclic group does notcontain any subgroup isomorphic to Dihn.The analogous pre-image construction, using Pin(2) instead of Pin(2), yields another dihedral group, Dihn, ratherthan a dicyclic group.

16.4 GeneralizationsLet A be an abelian group, having a specic element y in A with order 2. A group G is called a generalized dicyclicgroup, written as Dic(A, y), if it is generated by A and an additional element x, and in addition we have that [G:A] =2, x2 = y, and for all a in A, x1ax = a1.Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groupsare just a specic type of generalized dicyclic group.

16.5 See also binary polyhedral group binary cyclic group binary tetrahedral group binary octahedral group binary icosahedral group

16.6 References[1] Nicholson, W. Keith (1999). Introduction to Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. p. 449.

ISBN 0-471-33109-0.

[2] Roman, Steven (2011). Fundamentals of Group Theory: An Advanced Approach. Springer. pp. 347348. ISBN9780817683016.

Coxeter, H. S. M. (1974), 7.1 The Cyclic and Dicyclic groups, Regular Complex Polytopes, Cambridge Uni-versity Press, pp. 7475.

Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York:Springer-Verlag. ISBN 0-387-09212-9.

Chapter 17

Binary icosahedral group

In mathematics, the binary icosahedral group 2I or is a certain nonabelian group of order 120. It is anextension of the icosahedral group I or (2,3,5) of order 60 by a cyclic group of order 2, and is the preimage of theicosahedral group under the 2:1 covering homomorphism

Spin(3)! SO(3)

of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroupof Spin(3) of order 120.It should not be confused with the full icosahedral group, which is a dierent group of order 120, and is rather asubgroup of the orthogonal group O(3).The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, underthe isomorphism Spin(3) = Sp(1) where Sp(1) is the multiplicative group of unit quaternions. (For a description ofthis homomorphism see the article on quaternions and spatial rotations.)

17.1 ElementsExplicitly, the binary icosahedral group is given as the union of the 24 Hurwitz units

{ 1, i, j, k, ( 1 i j k ) }

with all 96 quaternions obtained from

( 0 i 1j k )

by an even permutation of all the four coordinates 0, 1, 1, , and with all possible sign combinations. Here = (1 + 5) is the golden ratio.In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unitquaternion group Sp(1). The convex hull of these 120 elements in 4-dimensional space form a regular 4-polytope,known as the 600-cell.

17.2 Properties

17.2.1 Central extensionThe binary icosahedral group, denoted by 2I, is the universal perfect central extension of the icosahedral group, andthus is quasisimple: it is a perfect central extension of a simple group.

35

36 CHAPTER 17. BINARY ICOSAHEDRAL GROUP

Explicitly, it ts into the short exact sequence

1! f1g ! 2I ! I ! 1:

This sequence does not split, meaning that 2I is not a semidirect product of { 1 } by I. In fact, there is no subgroupof 2I isomorphic to I.The center of 2I is the subgroup { 1 }, so that the inner automorphism group is isomorphic to I. The full automorphismgroup is isomorphic to S5 (the symmetric group on 5 letters), just as for I = A5 - any automorphism of 2I xes thenon-trivial element of the center (1 ), hence descends to an automorphism of I, and conversely, any automorphismof I lifts to an automorphism of 2I, since the lift of generators of I are generators of 2I (dierent lifts give the sameautomorphism).

17.2.2 SuperperfectThe binary icosahedral group is perfect, meaning that it is equal to its commutator subgroup. In fact, 2I is the uniqueperfect group of order 120. It follows that 2I is not solvable.Further, the binary icosahedral group is superperfect, meaning abstractly that its rst two group homology groupsvanish: H1(2I;Z) = H2(2I;Z) = 0: Concretely, this means that its abelianization is trivial (it has no non-trivialabelian quotients) and that its Schur multiplier is trivial (it has no non-trivial perfect central extensions). In fact, thebinary icosahedral group is the smallest (non-trivial) superperfect group.The binary icosahedral group is not acyclic, however, as Hn(2I,Z) is cyclic of order 120 for n = 4k+3, and trivial forn > 0 otherwise, (Adem & Milgram 1994, p. 279).

17.2.3 IsomorphismsConcretely, the binary icosahedral group is a subgroup of Spin(3), and covers the icosahedral group, which is asubgroup of SO(3). Abstractly, the icosahedral group is isomorphic to the symmetries of the 4-simplex, which is asubgroup of SO(4), and the binary icosahedral group is isomorphic to the double cover of this in Spin(4). Note that thesymmetric group S5 does have a 4-dimensional representation (its usual lowest-dimensional irreducible representationas the full symmetries of the (n 1) -simplex), and that the full symmetries of the 4-simplex are thus S5; not thefull icosahedral group (these are two dierent groups of order 120).The binary icosahedral group can be considered as the double cover of the alternating groupA5; denoted 2 A5 = 2I;this isomorphism covers the isomorphism of the icosahedral group with the alternating group A5 = I; and can bethought of as sitting as subgroups of Spin(4) and SO(4) (and inside the symmetric group S5 and either of its doublecovers 2 S5 ; in turn sitting inside either pin group and the orthogonal group Pin(4)! O(4) ).Unlike the icosahedral group, which is exceptional to 3 dimensions, these tetrahedral groups and alternating groups(and their double covers) exist in all higher dimensions.One can show that the binary icosahedral group is isomorphic to the special linear group SL(2,5) the group of all22 matrices over the nite eld F5 with unit determinant; this covers the exceptional isomorphism of I = A5 withthe projective special linear group PSL(2,5).Note also the exceptional isomorphism PGL(2; 5) = S5; which is a dierent group of order 120, with the commu-tative square of SL, GL, PSL, PGL being isomorphic to a commutative square of 2 A5; 2 S5; A5; S5; which areisomorphic to subgroups of the commutative square of Spin(4), Pin(4), SO(4), O(4).

17.2.4 PresentationThe group 2I has a presentation given by

hr; s; t j r2 = s3 = t5 = rsti

or equivalently,

17.3. RELATION TO 4-DIMENSIONAL SYMMETRY GROUPS 37

hs; t j (st)2 = s3 = t5i:Generators with these relations are given by

s = 12 (1 + i+ j + k) t =12 ('+ '

1i+ j):

17.2.5 SubgroupsThe only proper normal subgroup of 2I is the center { 1 }.By the third isomorphism theorem, there is a Galois connection between subgroups of 2I and subgroups of I, wherethe closure operator on subgroups of 2I is multiplication by { 1 }.1 is the only element of order 2, hence it is contained in all subgroups of even order: thus every subgroup of 2I iseither of odd order or is the preimage of a subgroup of I. Besides the cyclic groups generated by the various elements(which can have odd order), the only other subgroups of 2I (up to conjugation) are:

binary dihedral groups of orders 12 and 20 (covering the dihedral groups D3 and D5 in I). The quaternion group consisting of the 8 Lipschitz units forms a subgroup of index 15, which is also the dicyclicgroup Dic2; this covers the stabilizer of an edge.

The 24 Hurwitz units form an index 5 subgroup called the binary tetrahedral group; this covers a chiraltetrahedral group. This group is self-normalizing so its conjugacy class has 5 members (this gives a map2I ! S5 whose image is A5 ).

17.3 Relation to 4-dimensional symmetry groupsThe 4-dimensional analog of the icosahedral symmetry group I is the symmetry group of the 600-cell (also that ofits dual, the 120-cell). Just as the former is the Coxeter group of type H3, the latter is the Coxeter group of typeH4, also denoted [3,3,5]. Its rotational subgroup, denoted [3,3,5]+ is a group of order 7200 living in SO(4). SO(4)has a double cover called Spin(4) in much the same way that Spin(3) is the double cover of SO(3). Similar to theisomorphism Spin(3) = Sp(1), the group Spin(4) is isomorphic to Sp(1) Sp(1).The preimage of [3,3,5]+ in Spin(4) (a four-dimensional analogue of 2I) is precisely the product group 2I 2I oforder 14400. The rotational symmetry group of the 600-cell is then

[3,3,5]+ = ( 2I 2I ) / { 1 }.

Various other 4-dimensional symmetry groups can be constructed from 2I. For details, see (Conway and Smith, 2003).

17.4 ApplicationsThe coset space Spin(3) / 2I = S3 / 2I is a spherical 3-manifold called the Poincar homology sphere. It is an exampleof a homology sphere, i.e. a 3-manifold whose homology groups are identical to those of a 3-sphere. The fundamentalgroup of the Poincar sphere is isomorphic to the binary icosahedral group, as the Poincar sphere is the quotient ofa 3-sphere by the binary icosahedral group.

17.5 See also binary polyhedral group binary cyclic group

38 CHAPTER 17. BINARY ICOSAHEDRAL GROUP

binary dihedral group binary tetrahedral group binary octahedral group

17.6 References Adem, Alejandro; Milgram, R. James (1994), Cohomology of nite groups, Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences] 309, Berlin, New York: Springer-Verlag,ISBN 978-3-540-57025-7, MR 1317096

Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups, 4th edition. NewYork: Springer-Verlag. ISBN 0-387-09212-9. 6.5 The binary polyhedral groups, p. 68

Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters,Ltd. ISBN 1-56881-134-9.

Chapter 18

Binary octahedral group

In mathematics, the binary octahedral group, name as 2O or is a certain nonabelian group of order 48. Itis an extension of the octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage ofthe octahedral group under the 2:1 covering homomorphism Spin(3) ! SO(3) of the special orthogonal group bythe spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48.The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, underthe isomorphism Spin(3) = Sp(1) where Sp(1) is the multiplicative group of unit quaternions. (For a description ofthis homomorphism see the article on quaternions and spatial rotations.)

18.1 ElementsExplicitly, the binary octahedral group is given as the union of the 24 Hurwitz units

f1;i;j;k; 12 (1 i j k)gwith all 24 quaternions obtained from

1p2(1 1i+ 0j + 0k)

by a permutation of coordinates (all possible sign combinations). All 48 elements have absolute value 1 and thereforelie in the unit quaternion group Sp(1).

18.2 PropertiesThe binary octahedral group, denoted by 2O, ts into the short exact sequence

1! f1g ! 2O ! O ! 1:This sequence does not split, meaning that 2O is not a semidirect product of {1} by O. In fact, there is no subgroupof 2O isomorphic to O.The center of 2O is the subgroup {1}, so that the inner automorphism group is isomorphic toO. The full automorphismgroup is isomorphic to O Z2.

18.2.1 PresentationThe group 2O has a presentation given by

39

40 CHAPTER 18. BINARY OCTAHEDRAL GROUP

hr; s; t j r2 = s3 = t4 = rsti

or equivalently,

hs; t j (st)2 = s3 = t4i:

Generators with these relations are given by

s = 12 (1 + i+ j + k) t =1p2(1 + i):

18.2.2 SubgroupsThe quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6. The quotientgroup is isomorphic to S3 (the symmetric group on 3 letters). The binary tetrahedral group, consisting of the 24Hurwitz units, forms a normal subgroup of index 2. These two groups, together with the center {1}, are the onlynontrivial normal subgroups of 2O.The generalized quaternion group of order 16 also forms a subgroup of 2O. This subgroup is self-normalizing so itsconjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups of orders 8 and 12in 2O. All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).

18.3 Higher dimensionsThe binary octahedral group generalizes to higher dimensions: just as the octahedron generalizes to the hyperoctahedron,the octahedral group in SO(3) generalizes to the hyperoctahedral group in SO(n), which has a binary cover under themap Spin(n)! SO(n):

18.4 See also binary polyhedral group binary cyclic group binary dihedral group binary tetrahedral group binary icosahedral group hyperoctahedral group

18.5 References Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups, 4th edition. NewYork: Springer-Verlag. ISBN 0-387-09212-9.

Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters,Ltd. ISBN 1-56881-134-9.

Chapter 19

Binary polyhedral group

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin xed,or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of allisometries that leave the origin xed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroupof the Euclidean group E(3) of all isometries.Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possiblesymmetries. All isometries of a bounded 3D object have one or more common xed points. We choose the origin asone of them.The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group orproper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D spaceitself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of amolecule and of molecular orbitals forming covalent bonds, and in this context they are also calledmolecular pointgroups.Finite Coxeter groups are a special set of point groups generated purely by a set of reectional mirrors passing throughthe same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter-Dynkin diagram. Coxeternotation oers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and othersubsymmetry point groups.

19.1 Group structureSO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it containsthose that leave the origin xed.O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix I):

O(3) = SO(3) { I , I }

Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Alsothere is a 1-to-1 correspondence between all groups of direct isometries H in O(3) and all groups K of isometries inO(3) that contain inversion:

K = H { I , I }H = K SO(3)

For instance, if H is C2, then K is C, or if H is C3, then K is S6. (See lower down for the denitions of thesegroups.)If a group of direct isometries H has a subgroup L of index 2, then, apart from the corresponding group containinginversion there is also a corresponding group that contains indirect isometries but no inversion:

41

42 CHAPTER 19. BINARY POLYHEDRAL GROUP

M = L ( (H \ L) { I } )

where isometry ( A, I ) is identied with A. An example would be C4 for H and S4 for M.Thus M is obtained from H by inverting the isometries in H \ L. This group M is as abstract group isomorphic withH. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation groupby inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O(2,R) and SO(2,R).Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of thegroup of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations (Cn) is normalboth in the group obtained by adding reections in planes through the axis (Cnv) and in the group obtained by addinga reection plane perpendicular to the axis (Cnh).

19.2 3D isometries that leave origin xedThe isometries of R3 that leave the origin xed, forming the group O(3,R), can be categorized as follows:

SO(3,R):

identity rotation about an axis through the origin by an angle not equal to 180 rotation about an axis through the origin by an angle of 180

the same with inversion (x is mapped to x), i.e. respectively:

inversion rotation about an axis by an angle not equal to 180, combined with reection in the plane through theorigin perpendicular to the axis

reection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.See also the similar overview including translations.

19.3 ConjugacyWhen comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not havethe same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groupsare conjugate subgroups of O(3) (two subgroups H1, H2 of a group G are conjugate, if there exists g G such thatH1 = g1H2g ).For example two 3D objects have the same symmetry type:

if both have mirror symmetry, but with respect to a dierent mirror plane

if both have 3-fold rotational symmetry, but with respect to a dierent axis.

In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry typeif and only if there is a rotation mapping the whole structure of the rst symmetry group to that of the second. (Infact there will be more than one such rotation, but not an innite number as when there is only one mirror or axis.)The conjugacy denition would also allow a mirror image of the structure, but this is not needed, the structure itselfis achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two oppositedirections. (The structure is chiral for 11 pairs of space groups with a screw axis.)

19.4. INFINITE ISOMETRY GROUPS 43

19.4 Innite isometry groupsThere are many innite isometry groups; for example, the "cyclic group" (meaning that it is generated by one element not to be confused with a torsion group) generated by a rotation by an irrational number of turns about an axis. Wemay create non-cyclical abelian groups by adding more rotations around the same axis. There are also non-abeliangroups generated by rotations around dierent axes. These are usually (generically) free groups. They will be inniteunless the rotations are specially chosen.All the innite groups mentioned so far are not closed as topological subgroups of O(3). We now discuss topologicallyclosed subgroups of O(3).The whole O(3) is the symmetry group of spherical symmetry; SO(3) is the corresponding rotation group. The otherinnite isometry groups consist of all rotations about an axis through the origin, and those with additionally reectionin the planes through the axis, and/or reection in the plane through the origin, perpendicular to the axis. Those withreection in the planes through the axis, with or without reection in the plane through the origin perpendicular tothe axis, are the symmetry groups for the two types of cylindrical symmetry. Note that any physical object havinginnite rotational symmetry will also have the symmetry of mirror planes through the axis.See also rotational symmetry with respect to any angle.

19.5 Finite isometry groupsSymmetries in 3D that leave the origin xed are fully characterized by symmetries on a sphere centered at the origin.For nite 3D point groups, see also spherical symmetry groups.Up to conjugacy the set of nite 3D point groups consists of:

7 innite series with at most one more-than-2-fold rotation axis; they are the nite symmetry groups on aninnite cylinder, or equivalently, those on a nite cylinder. They are sometimes called the axial or prismaticpoint groups.

7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups withmultiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes thepossible combinations are:

4 3-fold axes 4 3-fold axes and 3 4-fold axes 10 3-fold axes and 6 5-fold axes

A selection of point groups is compatible with discrete translational symmetry: 27 from the 7 innite series, and 5of the 7 others, the 32 so-called crystallographic point groups. See also the crystallographic restriction theorem.

19.6 The seven innite series of axial groupsThe innite series of axial or prismatic groups have an index n, which can be any integer; in each series, the nthsymmetry group contains n-fold rotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle360/n. n=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotationalsymmetry (see cyclic symmetries) and three with additional axes of 2-fold symmetry (see dihedral symmetry). Theycan be understood as point groups in two dimensions extended with an axial coordinate and reections in it. They arerelated to the frieze groups;[1] they can be interpreted as frieze-group patterns repeated n times around a cylinder.The following table lists several notations for point groups: HermannMauguin notation (used in crystallography),Schnies notation (used to describe molecular symmetry), orbifold notation, and Coxeter notation. The latter threeare not only conveniently related to its properties, but also to the order of the group. It is a unied notation, alsoapplicable for wallpaper groups and frieze groups. The crystallographic groups have n restricted to 1, 2, 3, 4, and 6;removing crystallographic restriction allows any positive integer.The series are:

44 CHAPTER 19. BINARY POLYHEDRAL GROUP

For odd n we have Zn = Zn Z2 and Dihn = Dihn Z2.The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, thatcan be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).The simplest nontrivial ones have Involutional symmetry (abstract group Z2 ):

Ci inversion symmetry C2 2-fold rotational symmetry Cs reection symmetry, also called bilateral symmetry.

The second of these is the rst of the uniaxial groups (cyclic groups) Cn of order n (also applicable in 2D), which aregenerated by a single rotation of angle 360/n. In addition to this, one may add a mirror plane perpendicular to theaxis, giving the group Cnh of order 2n, or a set of n mirror planes containing the axis, giving the group Cnv, also oforder 2n. The latter is the symmetry group for a regular n-sided pyramid. A typical object with symmetry group Cnor Dn is a propeller.If both horizontal and vertical reection planes are added, their intersections give n axes of rotation through 180, sothe group is no longer uniaxial. This new group of order 4n is called Dnh. Its subgroup of rotations is the dihedralgroup Dn of order 2n, which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but nomirror planes. Note that in 2D Dn includes reections, which can also be viewed as ipping over at objects withoutdistinction of front- and backside, but in 3D the two operations are distinguished: the group contains ipping over,not reections.There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the mainrotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reection in thehorizontal plane and a rotation by an angle 180/n. Dnh is the symmetry group for a regular (n+2)-sided prisms andalso for a regular (2n)-sided bipyramid. Dnd is the symmetry group for a regular (n+2)-sided antiprism, and also fora regular (2n)-sided trapezohedron. Dn is the symmetry group of a partially rotated prism.The groups D2 and Dh are noteworthy in that there is no special rotation axis. Rather, there are three perpendicular2-fold axes. D2 is a subgroup of all the polyhedral symmetries (see below), and Dh is a subgroup of the polyhedralgroups T and O. D2 can occur in homotetramers such as Concanavalin A, in tetrahedral coordination compoundswith four identical chiral ligands, or in a molecule such as tetrakis(chlorouoromethyl)methane if all the chlorouo-romethyl groups have the same chirality. The elements of D2 are in 1-to-2 correspondence with the rotations givenby the unit Lipschitz quaternions.The group Sn is generated by the combination of a reection in the horizontal plane and a rotation by an angle 360/n.For n odd this is equal to the group generated by the two separately, Cnh of order 2n, and therefore the notation Snis not needed; however, for n even it is distinct, and of order n. Like Dnd it contains a number of imp