Journal of Geometry. Vol. 4/2 1974. Birkh~user Verlag Basel

PATHOLOGICAL PROJECTIVE PLANES: ASSOCIATE AFFINE PLANES

E. Mendelsohn

University of Toronto

In [7] the author showed the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes. Similar pathologies are obtainable with respect to collineation groups of associated affine planes. (i.e. the affine planes obtained by distin- guishing a line as the line at infinity) as expressable in the following theorem.

THEOREM i: Let G be a group then there exists a projective

plane p such that for every normal subgroup N of G (including

1 and G) there is a line s that if one takes s the line

at infinity,the collineation group of the affine plane, pz=~

is N.

The technique developed by author in [7] is basically

that of obtaining a graph with a given graph theoretical

property and using theory of categories to translate this

to a property of projective planes. Although these techniques

will be reviewed briefly here the author recommends reference

the [7] for complete detail.

Results needed from projective qeometry

We shall need the following well known results from

projective geometry:

THEOREM 2: The collineation group of ps is isomorphic

to the subgroup of the collineation group stabalizing s [5]

THEOREM 3: Given a partial plane p there exists a free

completion of this plane to a projective plane. [2]

THEOREM 4: If a partial is confined (i.e. contains at

least three points on each line and at least three lines

through every point) then the free completion has the same

collineation group as the partial plane and furthermore the

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the isomorphism is given by restriction. [2]

Preliminary results needed from combinatorial category

theory

DEFINITION: Let A, B be concrete categories with under-

lying set functions U and U'. F:A§ i_~s a ful___~l embedding by

extension if HomA(AI,A2) ~ HomB(F(AI),F(A2)) and ther_____~e

exist a natural transformation n:U+U'F such that

Uf UA > UB

U"P (f) U'F(A) >U'F(B) commutes and ~A_~S i-i

for all A.

This says that the morphisms i_~n F (A) in extend the motions

o_ff th__~e morphisms i__nn A.

THEOREM 5: Given a graph I (X,R) with no loops, isolated

points or two-cycles such that each point is related to at

least two others, there is a confined partial plane p' such

that the group of collineations of p' ~ Aut(X,R) and further-

more to each arrow (x,y)eR there is a line ~, such that

Stab (X,R) ~ to the stabulizer of Z in the collineations (x,y)

of p' [7]

THEOREM 6: R(2i)ic I can be fully embedded by extension

into R(2) by a functor F such that for all (x,y)ER i there

exists (x',y')eR, (F((X,Ri)=(Y,R)) such that

Stab(x , y.)(X Ri)ic I = Stab(x y) (Y,R) and (Y,R) satisfies

the hypotheses of theorem 5.

Proof: This is implicit in the embedding given in

[4] together with a simple application of the techniques

of [6].

The graph theoretic terms will be defined in the section on graph theory.

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MENDELSOHN 3

Results need from Graph Theory

DEFINITION: The c a t e g o r y o f I - m u l t i c o l o r e d g r a p h s

R(2i)iEi, I a set, is defined as follows

O b j e c t s : A s e t x t o c j e t h e r w i t h a f a m i l y o f s u b s e t s

R. cXxX iel. l

Morphisms: f: (X,R i) + (Y,S i) (iel) is a morphism if

f:X§ is a function and for all iEI (x,x')cR.=>(f(x),f(x')) 1

s 1

We shall refer to R(2i)ie{l } by R(2) and call it the

category of graphs. The elements of R. are called i-

(colored)-arrows. We shall say "f preserves R." if f:x§ 1

is a m o r p h i s m f r o m (X,R i ) t o (Y,S i ) i n R ( 2 ) . A l o o p o f

R i i s a p o i n t x , s u c h t h a t ( x , x ) cR i , an i s o l a t e d p o i n t

of R i is a point such that ~/y(x,y)~R. and (y,x)~Ri; a 1

two-cycle in R i is a pair (x,y) such that (x,y) and

(y,x) ~R.

Aut(X,Ri)ic I ={fJf and f-IE Hom (X,Ri)ic I, (X,Ri)ieI)};

-~ 1 . (X,Ri)ic I is rigid if Hom (X,Ri)ici(X,Ri)ie I x

Stab(x,y ) (X,Ri)ic I) = {flf Aut(X,Ri)ic I f(x)=x f(y)=y}

THEOREM 7: (Caley-Frucht) Let G be a group and

(X,Rg)geG be a graph with [G I colors defined by X=IGland -i

(gl'g2)c~ <==> gl h = g2" Then Aut(X,Rg)g~G = G [i].

To complete the proof of Theorem 1 we need only tie

together theorems 2-7 with the following.

LEMMA i- Let G be a group then there exists a multi-

colored graph (X,Ri)ie I such that

-~ G (i) Aut (X, Ri) ic I -

(ii) Fon N4G(including N=I and N=G) 3(x,y)cRi(N)

such that Stab(x,y ) ((X,Ri)ie I) ~ N

Proof: Let I = {a,b,c}U{GxN} where N ={NINAG,

(N=I and N=G included).

~NGIN~G} 6 {HgIHgEG, H~G} Let X = {_--_

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4 MENDELSOHN

R a = {a rigid graph on {GIN4G}}

G INg } R b = { (~, Ng)

R = {(Ng,Kh)I NgcKh and N~K} c

R(N,g) = { (Nh,Nh') I Nh=Nh'g}

G (Nh) =Ngh. Thus f If geG define fg by fg(G)=~, N<G, fg g

preserves R a, ~, Rc trivially and if (Nh,Nh')eR(N,s)

then fg(Nh)=Ngh and fg(Nh') = Ngh' = Nghs and (Ng h, Ng h,)

eR(N,s ) thus fgeAUt(X,Ri) i I and fgh = fgfh" We claim

further that feAut(X,R.). _ => f=f for some g G. As f G G 1 Is g G-

preserves R f(--) =•; as f preserves ~ f({N IN e=}c a ~ ~ D g g L~

{NgINgCN}. Now consider f on R(l,g ) geG. (l,g) ER(l,g)

thus (f(1), f(g))ER(l,g)SO f(g)=f(1)g. Thus

f G/<I > =f f(1) Now considering that f preserves R(N,g ), N~G, we see (N,Ng

eR(N,g ) thus (f(N), f(Ng))eRNg so f(Ng) = f(N)Ng on the G

elements of ~. Now (l,N)eR2,so (f(1)f(N))eR 2 thus

f(1)ef(N) thus f(N)=N f(1) so f(N)Ng = N(f(1))Ng = Nf(1)g

thus f=f and (i) is proved. g

Let N~G, we must find (x,y)~(i) such

N if N=G we choose any arrow of R Stab (x,y) (X'Ri) ieI " a

all of which are stabilized by every feAut(X,Ri)iE I. If

G~N then in (N,G)~. If fg stabilizes (N,G)----> fg(N)=N

and fg(G). But fg(N)=N ~-> geN thus Stab(G,N ) (X,Ri)iei~N.

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MENDELSOHN 5

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Hedrlin Z. and Pultr A, Vopenka P., A rigid relation exists on any set. Comment Math. V. Carolinae 6 (1965) 149-155

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Mendelsohn E. On a technique for representing semi- groups as endomorphism semigroups of graphs with given properties, Semigroup Forum vol. 4 (1972) 283-294

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E. Mendelsohn Department of Mathematics University of Toronto ~oronto M5S IAI Ont. Canada

(Eingegangen am 2o.6.1973)

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