Journal of Geometry. Vol. 2/2, 1972. Birkhiuser Verlag Basel

EVERY GROUP IS THE COLLINEATION GROUP

OF SOME PROJECTIVE PLANE

E. Mendelsohn

Introduction:

We shall use in this paper, graph theoretical results

and a process adopted from category theory to translate

these results into the following results of projective

geometry.

We shall prove the following three theorems.

THEOREM i. If G is a group, then there exists a

projective plane P = (P,L,I) such that the collineation

group of P , A(P) ~ G , and if ~ is a cardinal,

,Vo IPE = IGI

THEOREM 2. If G. , {i { I} is a well ordered -- 1

sequence of groups then there exist projective planes

Pi' such that P. is a subplane of P. whenever i < j l - - - - -- ]

and A(Pi ) ~ G i .

THEOREM 3. Let groups G 1 and G 2 be given. Then there

exist projective planes P1 = (PI'LI'II) and P2 =

(P2,L2,I2) and a pair of onto maps ~: P1 + P2 and

~: L 1 § L 2 such that Pill=> #(p) I ~(/) (an onto

homomorphism [5]) with A(PI ) ~ G 1 and ~(P2 ) ~ G 2 .

We shall use the following notations and definitions:

A configuration C = (P,L,I) will be a set of points

2 MENDELSOHN

P , and a set of lines L and a relation of incidence

I c p • L such that

(I) pi IZ j i,j = 1,2 => (Pl ~ P2 =

and ( Z 1 ~ 12 = Pl = P2 )

Z 1 = Z 2 )

(2) There exist 4 points no three of which are

collinear.

or (3) There exist 4 lines no three of which are

concurrent.

A configuration is finite <=> IPULI < X0 �9

configuration is said to be confined iff

A finite

(i) Every point lies on at least 3 lines.

(2) Each line has at least 3 points on it.

The configuration C = (P,L,I) is a sub-configuration

of C' = (P', L', I')<=> P c p, , L c L', (PxL) n I' = I .

An infinite configuration is confined iff each

point and each line is contained in a finite confined

subconfiguration. The free completion of a configuration

to a projective plane is shown to exist in CI~ , and

will be denoted F( C )

By an onto homomorphism of C to C' we shall mean

a pair of onto maps, ~: P § P' and ~: L + L' such

that pIZ = r ~( Z ). An onto homomorphism from

C to C whose inverse is also an onto homorphism is

called a collineation, the group of all collineations of a

configuration will be denoted by A(C ) (The pre-

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MENDELSOHN 3

ceding definitions will be used for projective planes as

well as configurations.)

We shall make use of the following slight generaliza-

tion of a lemma found in ~i~ .

LEMMA i. Ever Z collineation o_~f C can be extended

t__oo collineation of F( C ) and when C i_~s confined every

collineation of F(C ) restricts to a ccilineation of C

A graph shall be a pair (X,R) , R c X x X A

map of graphs ~ : (X,R) + (Y,X) will be a map 2 2

: X § Y such that ~ (R) c S where ~ (x,y) =

[ ~ (x), ~ (y)] A mapping ~ : (X,R) § (X,R) is an

automorphism iff ~ is bijective and ~-i is a map

of graphs. We let A(X~R) be the group of automorphisms 2

of (X,R) . A map is arrow-onto if ~ : R § S is

onto. We say there is an arrow from x to y in

(X,R) <=>(x,y) ~ R . We say there is a loop at x if

(x,x) e R; a point x is isolated if there is no arrow

coming from or ending up at it; a two-cycle is a pair

(x,y) such that (x,y) and (y,x) r R . We shall use

the following theorems from graph theory which sre to be

found in the reference immediately following each

statement :

THEOREM 4. If G is a r ~ then there exists a

graph (X,R) such that A(X,R) ~ G, and if ~ >- ~ o

is a cardinal Ixl = IGl E33.Ecj

THEOREM 5. If G i , i e I is a well ordered

s_e_guence of r o ~ then there exist _g_[_aphs (Xi,R i) with

. c R i < j and A(Xi,Ri ) =~ G i . E4~ and Xi c Xj , R 1 3 ~ ' .....

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4 MENDELSOHN

THEOREM 6.

graphs (X 1 , R 1 )

: (x I.R l) §

A (X 2,R 2 ) ~ 0 2 .

If GI,G 2 ar__ee rgr_o_u_ps the n there exist

and (X2,R 2) , and a__nn arrow-onto map

(X2,R2) such that A(XI,RI) ~ G 1 and

E 2 0 .

Furthermore by a technique found in E2] , E 4] and

E 6] one can assume that all graphs which appear in the

above theorems have no loops, isolated points or two-

cycles, and that each point is related to at least 2

others. These assumptions are for technical reasons

which will reveal themselves in the particular construc-

tion and proof we shall use.

AS pecial Configuratio_n:

(The author wishes to thank J. Sichler at this

point for his suggestion that simpler and more palatable

configuration to the one originally proposed by the

author might be found, and to N.S. Mendelsohn who

pointed out that a configuration in the unpublished work

of S. Ditor might have all the properties desired. It

is the configuration of Ditor which is now used here.)

It is the configuration given by the following table,

the columns representing the lines (see Diag. i), the

letters points.

AC EA BDDC E A B

B N O F K N O K M K N

C L L G L K M G G O O

D HF HMF H

E

We shall call this configuration D = (PD,LD,ID)

IOO

M ENDE LS OHN

H

O

A B C 0 s

LEMMA i. ~ has no collineations other than the

identit Z.

Let ( ~ , ~ ) be any collineation of the configura-

tion provided by the above table in which the points are

A, B, C, etc., and the lines are the columns of the

table. Since K and O are the only 4-points, they

must either be fixed or map into each other. Then KO

and A are fixed so that AF, a 4-1ine, and AB, a 5-

line, are each fixed. Hence G on AF is fixed, no

other point on AF being on two 3-1ines, and D on

AB is fixed, no other point on AB being on two 4-

lines. It follows that GK and GM are each fixed

since GK and the fixed line AK intersect whereas GM

and AK do not intersect. Then K, C, and E are fixed,

being points of intersection of fixed lines; 0 is

fixed since A and K are each fixed; B is fixed

since A and K are each fixed; B is fixed since

every other point on AB is fixed; N is fixed since

O and B are each fixed. Now there are at least two

fixed points on every line so that every line, and hence

every point, must be fixed and ( 4 , ~ ) must be the

identity collineation.

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6 MENDELSOHN

The Pastin~ of D on a Gra~:

The process we shall use is analogous to the sip

process ~6~ . Let G = (X,R) be a graph without loops,

isolated points or two-cycles and with each point

connected to at least 6 others. We define the configura-

tion (G) in the following way:

r_ The points of ~(G): P(G) = It{PD

The lines of D(G): L(G) = L D x R

The Incidence of ~(G): ]l G by:

(i) (p,r) I G ( Z ,s) <=> r = s and PIDs

x I G (Z,s) <=> s = (x,y) , KIDZ

y I G (Z,s) <=> s = (y,z) , OIDl

(ii)

(iii)

]0x}

Intuitively, we place a copy of ~ in place of every

arrow of R pasting the points K and/or O together

whenever they correspond. If we represent the con-

figuration V as in diagram 2 and paste it on the graph

given diagram 3 we get the configuration of Diagram 4.

D

K 0

/

Diagram 2. Diagram 3.

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MENDELSOHN 7

D IAGRAM 4.

THEOREM 7. If G = (X,R) is a graph such that

has n__oo loops, isolated points a n~ every point is

connected to at least 2 others then:

G

i. ~ (G) is a confined configuration.

2. There is an isomorphism between the collineation

group of D (G) and the automorphism group of the graph

G .

3. Ther___~eisa on___ee-on__eecorrespondence be___tween onto

maps from G to G* (where G* also has the properties

~iven in the ~ ) and onto homomorphism from D(G)

to ~(G*) .

PROOF: (i) is obvious, (3) is similar to (2). The

proof of (2) is as follows:

Let f: G + G be an automorphism of G, define

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8 MENDELSOHN

~(f) = (%f ,~ f) , the following way: ~f(x) = f(x) ,

%f(p,r) = (p,s) where r = (x,y) and s = If(x) ,f(y)] ;

~f(/,r) = (/,s) where 1 = (x,y) and s = [f(x) ,f(y)]

It is clear that }f is a collineation, } is one to one

and % is a homomorphism of Aut (G) into A[D (G) ] We

must show ~ is onto. Let (~,~) be a collineation of

DiG) . As the only points of ~(G) which are on at least

2 • 4 = 8 lines are those of X , %(X) c X , call

~I X = f ; furthermore no other points of D(G) map to

X as % is an automorphism, Now consider the points of

the form (p,r) for fixed r if }(p,r) = (q,s) and

$(p',r) = (q',s') , s' @ s then the line 1 = (p,r) A(p',r')

must map to a line l' �9 DiG) such that (q,s)I(G)/ and

(q',s')IiG)l . But there are no such lines in L(G)

Therefore s = s' Now by an argument similar to that of

lemma 1,

r u(x,y}Ir-- (x,y), pe PD- {k,0}}) c

r = [f(x),f(y)], p e PD-{k'0}}l

and furthermore ~(p,r) = (p,s) ~ As (x,y) �9 R

[f(x),f(y)] e R , f is a graph automorphism and

~f = ~ . It is easy to see that this now implies

~f = ~ , and thus (~,~) = #(f)

implies

THEOREM i. If G is a @roup then there exists a

projective plane P = (P,L,I) such that the collineation

group __~ P , A(P) =~ G , and if ~ a cardinal ~ -> ~0

PROOF: By theorem 4 there exists a graph (X,R)

without loops, isolated points or two-cycles and with

each point connected to at least 2 others with IXI = elGI-

Consider F(~(G)) iwhere F is the free completion).

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MENDELSOHN 9

A[F(~(G))] 2 G as r is confined, and if

F(~(G)) = (P,L,I) then IP = ,~0 IXl = ~IGI

THEOREM 2. If G i , 1 c I is a well ordered

sequence of groups then there exist projective planes

Pi ' such that P. is a subplane of P whenever i < j

and A(P i) ~ G i

PROOF: Similar to theorem 1 using theorem 5.

THEOREM 3. Let G i and G 2 b_~e given. Then there

exists projective planes P1 and (PI,LI,II) and

P2 = (P2'L2'I2) and a pair of onto maps r : P1 § P2

and Y : L 1 § L 2 such that pI/ => ~(p) I'~(/) , with

A(P I) ~ G 1 , and A(P 2) ~ G 2 .

PROOF: Similar to theorem 1 using theorem 6.

REFERENCES

i.

2.

3.

4.

5.

Hall, M.: Projective Planes. Trans. Amer. Math. Soc. 54(1943), 229-277.

Hedrlin, Z.: On Endomorphisms of Graphs and their Homomorphic Images. Proof Techniques in Graph Theory, Ed. F. Harrary, Academic Press (1969).

Hedrlin, Z. and Lambek, J.: How Comprehensive is the Category of Semigroups, A Journal of Algebra, Vol. ii, No. 2 (1969), 195-212.

Hedrlin, Z. and Mendelsohn, E.: The Category of Graphs with Given Subgraph. Can. J. Math. Vol. 21, No. 6 (1969), 1506-1517.

Hughes, D. R.: On Homomorphism of Projective Planes. Proc. Symp. Appl. Math. 10(1960), 45-52.

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6. Mendelsohn, E.: On a Technique for representing

semigroups as endomorphism semigroups of

graphs with given properties.

Semigroup Forum (To Appear).

E. Mendelsohn

Department of Mathematics

University of Toronto

Toronto 181, Ontario, Canada.

(Received November 25, 1971)

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