Pathological projective planes: Associate affine planes

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 distinguishing 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 1 and G) there is a line s at infinity,the

collineation

(including

that if one takes s the line 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 a property of projective planes.

to translate

Although

these techniques

will be reviewed briefly here the author recommends the

this

reference

[7] for complete detail.

Results needed f r o m

projective qeometry

We shall need the following well known results projective

from

geometry:

THEOREM 2: The collineation group of ps to the subgroup of the collineation THEOREM

group stabalizing

3: Given a partial plane p

there exists

completion of this plane to a projective plane. THEOREM

4:

is isomorphic

If a partial is confined

(i.e.

[5]

a free

[2] contains

least three points on each line and at least three through every point)

s

at

lines

then the free completion has the same

collineation group as the partial plane and furthermore

161

the

2

MENDELSOHN

the isomorphism is given by restriction. Preliminary

[2]

results needed from combinatorial

category

theory DEFINITION:

Let A, B be concrete categories with under-

lying set functions U and U'. extension if

HomA(AI,A2)

F:A§

i_~sa ful___~lembedding by

~ HomB(F(AI),F(A2))

exist a natural transformation

and ther_____~e

n:U+U'F such that

Uf UA

> UB

U"P (f) >U'F(B)

U'F(A)

commutes and ~A_~S i-i for all A.

This says that the morphisms o_ff th__~emorphisms THEOREM

i_~n F (A) in extend the motions

i__nnA.

5: Given

points or two-cycles least two others,

a graph I (X,R) with no loops, isolated such that each point is related to at

there is a confined partial plane p' such

that the group of collineations of p' ~ Aut(X,R) more to each arrow

and further-

(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

into R(2) by a functor exists

(x',y')eR,

fully embedded by extension

F such that for all

(F((X,Ri)=(Y,R))

(x,y)ER i there

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.

162

MENDELSOHN

Results

need

from Graph Theory The c a t e g o r y

DEFINITION: R(2i)iEi,

A set

R. cXxX iel. l Morphisms: s

of

I-multicolored

graphs

I a set, is defined as follows

Objects:

f:X§

3

x tocjether

with

a family

f: (X,R i) + (Y,S i)

is a function

of subsets

(iel) is a m o r p h i s m if

and for all iEI

(x,x')cR.=>(f(x),f(x')) 1

1 We shall refer to R(2i)ie{l } by R(2)

category

of graphs.

(colored)-arrows. from

Ri

x,

a point

(X,R i )

such

to

that

of R i is a point such that two-cycle

of R. are called i-

We shall say "f preserves

is a m o r p h i s m is

The elements

and call it the

in R i is a pair

(Y,S i )

in

( x , x ) cR i ,

R(2).

R." if f:x§ 1 A loop of

an i s o l a t e d

point

~/y(x,y)~R. and (y,x)~Ri; 1 (x,y) such that (x,y) and

a

(y,x) ~R. Aut(X,Ri)ic I ={fJf and f-IE Hom (X,Ri)ic I is rigid if Hom

(X,Ri)ic I, (X,Ri)ieI)}; (X,Ri)ici(X,Ri)ie I -~ 1 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 ( g l ' g 2 ) c ~ gl h = g2" Then A u t ( X , R g ) 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 colored graph

a multi-

(X,Ri)ie I such that

(i) Aut (X, Ri) ic I -~ - G (ii) Fon N 4 G ( i n c l u d i n g

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=I and N=G included). ~NGIN~G} 6 {HgIHgEG, H~G} Let X = {_--_

163

N ={NINAG,

4

MENDELSOHN

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

G Ng) R b = { (~, R

c

INg }

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

R(N,g) = { (Nh,Nh') I Nh=Nh'g} G N fg(N)=N and fg(G).

But fg(N)=N

~->

geN thus Stab(G,N ) (X,Ri)iei~N.

164

MENDELSOHN

5

References l,

Frucht, Herstellung von Graphen mit vorgegeben abstrakter Gruppe. Composito Math. 6 (1938) 239-250

2.

Hall N. Projective Planes, 54(1943) 229-277

3.

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

Trans. Amer.

Math.

Soc.

.

Hedrlin Z. and Lambek J, How comprehensive is the category of semigroups? Journal of Algebra, Vol ii No.2 (1969) 195-212

.

Lingenberg R. Grundlagen den Geometry graphiche Institute (1969) ZUrich

6.

.

i, Biblio-

Mendelsohn E. On a technique for representing semigroups as endomorphism semigroups of graphs with given properties, Semigroup Forum vol. 4 (1972) 283-294 M e n d o l s o h n E. Every group is the collineation group of some projective plane. Journal of Geometry Vol 2/2 (1972) 97-105.

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

(Eingegangen am 2o.6.1973)

165