Projective Planes with a Condition on Projectivities

Projective Planes with a Condition on Projectivities

:PETER LORIMEa

University of Auckland, Auckland, New Zealand 1

Introduction

L e t ~ be a projective plane, L, M two lines of ~, ~ the set of points on L excluding the point of intersection of L and M, a n d I a point of ~ lying on neither L nor M. I f J is a n y point of ~ n o t lying on L or M we can define a p e r m u t a t i o n r of ~ b y first projecting L onto M t h r o u g h I and t h e n pro- jecting M back onto L t h r o u g h J . We denote the set of p e r m u t a t i o n s r obtained from e v e r y such J b y B. ~ has t h e following properties which are characteristic of projective planes.

I I ~ R

I I I f ~,fi, y , $ E ~ , ~ # fi, y # ~ there is a u n i q u e m e m b e r r of R with the properties r(~) -~ 7, r(fi) = (~

I I I The relation ~ on J~ defined b y r ~ s if r = s or r ( c r s(c~) for e v e r y a C ~ , is an equivalence relation. E a c h equivalence class is sharply transitive on

~ , i.e. if a, f i e ~ each class contains e x a c t l y one m e m b e r r with r(~) = ft.

A set ~ of p e r m u t a t i o n s on a set ~ which satisfies I, I I and I I I will be called sharply d o u b l y transitive and if G is a group of p e r m u t a t i o n s on ~ which contains B we call R a sharply d o u b l y t r a n s i t i v e subset of G.

A consequence of the axiom of :pappus in a projective plane is t h a t a n y pro- j e c t i v i t y of a line which transposes two points o f t h e line must act as an involution on t h e line. The converse of this result follows from a t h e o r e m of J. Tits [4]. The unrestricted group of projectivities on a line in a projective plane acts t r i p l y t r a n s i t i v e l y on the points of t h a t line and it is a simple consequence of the result of Tits t h a t a t r i p l y transitive group in which only involutions transpose symbols must be a group P G L ( 2 , F ) of all bilinear t r a n s f o r m a t i o n s

1 This research was partially supported by :National Science Foundation grant GP-11342 at the University of Notre Dame.

1 4 0 I~E TEI~ L O R I M E ~

ax -~ b

x ~ - ad -- bc =/= 0 cx §

over a field F. This is enough to ensure t h a t Pappus' Theorem holds in the plane.

Here we investigate the less restrictive situation suggested in the first paragraph.

Our main result is

THEO~n~ 1. Let G be a group of permutations on a finite set ~ . Suppose that G contains a sharply doubly transitive subset 17 and satisfies

IV I f g E G, a E ~ , g(a) :/: a but g~(a) = a then g2 _~ 1.

Then G has a normal regular subgroup V and

(i) if a E ~ the stabilizer of a in G contains a subset 17, such that 17 ~ V1?, (ii) the stabilizer in G of any two elements of ~ is an elementary abelian 2-group

(possibly 1).

The proof relies heavily on M. Aschbacher's classification of finite doubly transitive groups in which the stabilizer of a n y two points is abelian.

Our result has the following consequence for the t h e o r y of projective planes T~EORE~ 2. Let ~ be a finite projective plane and let L, M, ~ , I and tt be as in the first paragraph above. I f 17 generates the subgroup G of the symmetric group on ~ and G satisfies I V then ~ is the dual of a translation plane.

The proof is direct and will n o t be given here.

Proo! of Theorem 1

We now suppose t h a t G satisfies the hypotheses of Theorem 1. I f ~, fi E and K is a n y subgroup of G we write K~ for the stabilizer of ~ in K and K ~ for the stabilizer of cr and fl in K.

We note first t h a t G is clearly doubly transitive on ~ .

L e t ~, fi be a n y two fixed symbols of ~ . As G is doubly transitive G contains a member a which interchanges ~ and ft. :By assumption we t h e n have a 2 --~ 1.

I f h

E G.a,

ah also interchanges a and fi so t h a t (ah)~ = 1. Hence every member of aGaa is an involution and so G~a is abelian. We can now apply Aschbacher's result [1] and deduce t h a t one of the following is true

(1) G has a normal regular subgroup.

(2) G : L 3 ( 2 ).

(3) G ~--17(3) the smallest l~ee group.

(4) G has a unique minimal normal subgroup M(G) with G C _ A u t M ( G ) and M ( G ) = L2(q) , U3(q), Sz(q) or 1?(q).

P R O J E C T I V E PLAI~*ES "vVITtI A C O N D I T I O I ~ - O N P R O J E C T I V I T I E S 1 4 1

B y e q u a l i t y in each case we m e a n t h a t G or

M(G)

is p e r m u t a t i o n i s o m o r p h i c t o t h e g i v e n g r o u p w h e r e t h e l a t t e r h a s its u s u a l r e p r e s e n t a t i o n as a d o u b l y t r a n s i t i v e p e r m u t a t i o n group. I-Iowever t h e r e is n o loss in g e n e r a l i t y b y a s s u m i n g t h a t this r e l a t i o n is e q u a l i t y .

Our i m m e d i a t e t a s k n o w is to show t h a t conclusions (2), (3) a n d (4) c a n n o t h o l d unless (1) also holds.

L3(2 ) contains t h e m a t r i x

tili) oo

w h i c h i n t e r c h a n g e s t h e v e c t o r s

a n d

b u t is n o t a n i n v o l u t i o n . T h u s G v~ L3(2 ).

_R(3) has o r d e r 28" 27 9 2 a n d degree 28. E v e r y i n v o l u t i o n of R(3) fixes f o u r symbols. S u p p o s e t h a t G ~ _R(3). R c o n t a i n s a m e m b e r i n t e r c h a n g i n g t w o s y m b o l s o f ~ a n d r 2 ~ 1. T h u s r fixes f o u r s y m b o l s o f ~ w h i c h is impossible as 1 is t h e o n l y m e m b e r o f r f i x i n g t w o s y m b o l s o f ~ . }fence G :# R(3).

N o w suppose t h a t case (4) of A s c h b a c h e r ' s r e s u l t applies. W e n o t i c e f i r s t t h a t in each case

M(G)

is d o u b l y t r a n s i t i v e on ~ . I n e a c h ease t h e stabilizer

M(G)~,

o f a in

M(G)

has a c h a r a c t e r i s t i c s u b g r o u p , s a y Q~ w h i c h is r e g u l a r on ~ - - {a}.

As

M(G)

is d o u b l y t r a n s i t i v e on ~ we h a v e

G ~ G~M(G).

I f a is a m e m b e r of G w h i c h i n t e r c h a n g e s ~ a n d fi each coset o f

M(G)

in G c o n t a i n s a m e m b e r of

aG~. As

all t h e m e m b e r s of this set are i n v o l u t i o n s it follows t h a t

G/M(G)

is a n e l e m e n t a r y a b e l i a n 2-group.

W e n o w consider t h e d i f f e r e n t possibilities for

M(G)

in t u r n .

S u p p o s e t h a t

M ( G ) ~ Le(q).

L e t x be a n y m e m b e r o f

G~. Then x

n o r m a l i s e s Q~. S u p p o s e h E Q - {1}. l~rom t h e p r o p e r t i e s o f A u t

L2(q)

t h e r e is a m e m b e r y o f

PGL(2, q)~

w i t h t h e p r o p e r t y h * : h r . P u t

z ~ xy -1

so t h a t h * : h. N o w let w be a n y m e m b e r o f

L2(q)~.

As b o t h G ~ a n d

PGL(2, q)~

are a b e l i a n a n d c o n t a i n

Le(q)~z

we h a v e (h~) = :

h w. L2(q)~a

contains e i t h e r q - 1 or 8 9 1) m e m b e r s a n d n o n e o f t h e s e c o m m u t e s w i t h a m e m b e r of O~. H e n c e z centralizes e i t h e r q or 89 ~- 1) m e m b e r s of Q~. As 89 -{- 1) ~> 89 we f i n d t h a t in e i t h e r case z centralizes Q~. As z centralizes

L2(q)~

it n o w follows t h a t z centralizes

L2(q) ~. G/M(G)

is a n e l e m e n t a r y a b e l i a n 2-group so t h a t

z 2 E M(G).

T h u s z ~ lies in t h e c e n t r e of

L2(q)~,

a n d so z 2 ~ 1. L e t a be a m e m b e r of L2(q ) which inter- changes ~ a n d

fi.

T h e n

az

also i n t e r c h a n g e s ~ a n d fl so we h a v e b o t h a 2 : 1 a n d

(az)2:

1. T h u s z centralizes a. As L2(q) ~ is a m a x i m a l s u b g r o u p of

L2(q)

it n o w follows t h a t z centralizes

L~(q).

B u t z E A u t

L2(q)

so t h a t z ~--1. H e n c e

142 P E T E R L O R I M E R

x E .PGL(2, q)

a n d so e i t h e r

G =-L~(q)

or

PGL(2, q).

I n e i t h e r case we get a c o n t r a d i c t i o n to t h e r e s u l t in [3] unless

G ~ L2(q)

w h e r e q ---- 2 or 3. B o t h o f t h e s e groups h a v e a n o r m a l r e g u l a r s u b g r o u p .

I n considering t h e g r o u p s

Ua(q)

we will a d o p t t h e n o t a t i o n o f [2, p. 233 f]. I f /c lies in t h e field w i t h q~ m e m b e r s

SU(3, q2)

c o n t a i n s t h e m a t r i x

w h i c h i n t e r c h a n g e s t h e subspaces ~wl} a n d ~wa}. T h e s q u a r e of this m a t r i x is

a n d t h e c o n d i t i o n s t h a t this be a scalar m a t r i x is /c 3(~-1) = 1, or k 3(~-1) = 1. This is t r u e w h e n k is a p r i m i t i v e e l e m e n t of t h e f i e l d o n l y w h e n q = 2. H e n c e we c a n n o t h a v e

M(G) = U3(q)

e x c e p t in t h e case q = 2. T h e n

U3(q)

is a :Frobenius g r o u p a n d G w o u l d h a v e a n o r m a l r e g u l a r s u b g r o u p .

S u p p o s e n e x t t h a t .M(G) =

Sz(q).

T h e n

M(G)~

has o d d o r d e r a n d as

G/M(G)

is an e l e m e n t a r y a b e l i a n 2-group e a c h eoset o f 2~F/(G)~ in G ~ c o n t a i n s e x a c t l y one i n v o l u t i o n and t h e y all lie in d i f f e r e n t classes o f G. L e t z be a n i n v o l u t i o n o f

G~--M(G)~r

a n d p u t D = { o E ~ . ; z ( ~ ) = o } . I f a, b E D we can easily show t h a t

c(z) = i x E a; x(.,) e x(b) C

This is b e c a u s e all t h e i n v o l u t i o n s o f G ~ lie in d i f f e r e n t classes o f G. H e n c e

C(z)

is d o u b l y t r a n s i t i v e on /2. N o w z E G~ so t h a t z n o r m a l i z e s Q~ a n d so n o r m a l i z e s

Z(Q~).

As

Z(Q~)

is a 2-group a n d z 2 ~ - 1, h ~ : h for some

h E Z ( Q ~ ) -

{1}. I f w E M ( G ) ~ ,

wz -~ zw

so t h a t (h~) ~ -~ h ~. :From t h e p r o p e r t i e s o f

Sz(q)

it follows t h a t z centralizes

Z(Q~).

N o w

Z(Q~)

has o r d e r q a n d Q~ has o r d e r q~. W e m a y suppose t h a t z centralizes

mq

m e m b e r s of Q~ w h e r e m divides q. z centralizes t h e q - - 1 m e m b e r s o f

M(G)~z.

H e n c e , b e c a u s e of t h e d o u b l e t r a n s i t i v i t y p r o p e r t y o f

C(z), z

centralizes

( q - - 1 ) m q ( m q ~

1) m e m b e r s o f

Sz(q).

This n u m b e r m u s t t h u s be a divisor o f t h e o r d e r o f

Sz(q)

w h i c h is (q - - 1)q2(q 2 -~ 1).

B e c a u s e m a n d q are powers o f 2 we can d e d u c e t h a t

mq-~

1 divides q2 ~ 1.

:From this we easily d e d u c e t h a t m --~ q a n d so z centralizes

Sz(q).

As z E A u t

Sz(q)

we m u s t h a v e z : 1, a c o n t r a d i c t i o n . T h u s

G ~-- M(G) -~ Sz(q)

w h i c h c o n t r a d i c t s t h e r e s u l t in [3]. H e n c e we c a n n o t h a v e

M(G) : Sz(q).

L a s t l y u n d e r (4) we consider t h e p o s s i b i l i t y t h a t

M(G)

--~/~(q), a g r o u p o f R e e t y p e . T h e p r o p e r t i e s o f these g r o u p s m a y be f o u n d in [5]. L e t t be t h e u n i q u e i n v o l u t i o n o f ~R(q)~. As

M ( G ) ~ G~, t

has no o t h e r c o n j u g a t e in G~. L e t ~2 be t h e subset of ~ f i x e d p o i n t w i s e b y t. As a b o v e we f i n d t h a t i f a, b E ~9, t h e n

P R O J E C T I V E P L A N E S W I T H A C O N D I T I O N O N P R O J E C T I V I T I E S 143 C(t) : {x E G; x(a) E .(2, x(b) E/2}.

H e n c e C(t) is d o u b l y t r a n s i t i v e on /2. As C(t) N R ( q ) = { 1 , t } x K w h e r e K is i s o m o r p h i c t o L~(q) we f i n d t h a t /2 m u s t h a v e q ~- 1 m e m b e r s a n d w e m u s t h a v e K p e r m u t a t i o n i s o m o r p h i c t o L~(q) as a p e r m u t a t i o n g r o u p on /2. W e w r i t e K = L2(q). S u p p o s e n o w t h a t z is a 2 - e l e m e n t o f G ~ z - M(G)~z. T h e n z ~ E M ( G ) ~ os t h a t e i t h e r z 2 - - - 1 or z~---t. I n e i t h e r case z ~ centralizes K . C l e a r l y z n o r m a l i z e s K . U s i n g t h e s a m e c o n s i d e r a t i o n s as we a p p l i e d a b o v e to t h e case M ( G ) ~ L~(q) we c a n p r o v e t h a t e i t h e r z c e n t r a l i z e s K or a c t s o n K ~ L2(q) as a n i n v o l u t i o n o f PGL(2, q)~. N o w consider t h e r e p r e s e n t a t i o n o f

C(t) as a p e r m u t a t i o n g r o u p on /2. F r o m t h e a b o v e it follows t h a t t h e i m a g e o f C(t) u n d e r this r e p r e s e n t a t i o n is e i t h e r L2(q) or PGL(2, q). T h e i m a g e o f R rl C(t) is t h e n a d o u b l y t r a n s i t i v e s u b s e t o f t h e g r o u p w h i c h occurs. T h i s c o n t r a d i c t s t h e r e s u l t o f [3] e x c e p t w h e n q = 3 a n d t h e g r o u p w h i c h occurs is L2(q). I n t h i s case we g e t t h e o n l y g r o u p of l%ee t y p e f o r t h e p a r a m e t e r q ~ 3, n a m e l y t h e R e e g r o u p R(3) o f o r d e r 28 9 27 9 2 a n d d e g r e e 28. B u t _R(3) _~ PILL(2, 8) _~ A u t L~(8). H e n c e A u t _ R ( 3 ) = _R(3) so t h a t G = M ( G ) ~ R(3) a n d we h a v e a l r e a d y s h o w n t h a t we c a n n o t h a v e G = R(3).

W e h a v e n o w s h o w n t h a t conclusions (2), (3) a n d (4) o f A s c h b a c h e r ' s r e s u l t c a n n o t h o l d unless (1) h o l d s also. W e n o w i n v e s t i g a t e conclusion (1) in m o r e detail.

A s s u m e f r o m n o w on t h a t G h a s a r e g u l a r n o r m a l s u b g r o u p V a n d let a b e a f i x e d m e m b e r o f ~ . I f H is t h e s t a b i l i z e r o f a in G we h a v e G = V H a n d V N H = 1. As G is d o u b l y t r a n s i t i v e V is a n e l e m e n t a r y a b e l i a n g r o u p a n d we m a y t a k e V as a v e c t o r s p a c e o v e r a p r i m e f i e l d 2, of o r d e r s a y p a n d H as a g r o u p o f l i n e a r t r a n s f o r m a t i o n s o v e r V. I f v E V, h E H we will d e n o t e t h e e l e m e n t vh o f G f r o m n o w on also b y (v, h). W e m a y i d e n t i f y t h e e l e m e n t s o f w i t h t h o s e o f V in s u c h a w a y t h a t w e m a y a s s u m e t h a t G is d o u b l y t r a n s i t i v e o n V a n d (v, h) is t h e m a p p i n g V --> V g i v e n b y w --> v ~ h(w). W e shall w r i t e (v, h)w ~ v -}- h(w). U n d e r t h i s i d e n t i f i c a t i o n a c o r r e s p o n d s t o 0.

S u p p o s e v E V - - {0}. T h e m e m b e r s o f G w h i c h i n t e r c h a n g e 0 a n d v a r e t h e e l e m e n t s (v, h) w i t h h(v) ~ v ~ O. S u c h a n e l e m e n t o f G m u s t b e a n i n v o l u t i o n a n d t h e c o n d i t i o n for t h i s is h ~ = 1.

W e n o w split o u r c o n s i d e r a t i o n s i n t o t w o p a r t s d e p e n d i n g on w h e t h e r p is o d d or even.

F i r s t s u p p o s e t h a t p is odd. L e t S b e a n y s h a r p l y d o u b l y t r a n s i t i v e s u b s e t o f G. T h e n S c o n t a i n s a m e m b e r (v, h), h ~ = 1 w h i c h i n t e r c h a n g e s 0 a n d v r 0.

I f h(w) = w for s o m e w E V - - { 0 } we h a v e

(v, h)( 89 + ~,w) = i v + ~w

f o r e a c h cr E 2,. B u t 1 is t h e o n l y m e m b e r o f S w h i c h f i x e s t w o m e m b e r s o f V so t h a t n o such w c a n exist. B u t H is a g r o u p o f l i n e a r t r a n s f o r m a t i o n s on V, h ~ = 1 a n d c h a r 2' :~ 2. T h e o n l y p o s s i b i l i t y is h = - 1. H e n c e S c o n t a i n s all

144 ~ E T E R L O R I ~ E ~

t h e v e c t o r s ( v , - - 1 ) , v r 0. S u p p o s e v r 0. T h e n t h e set ( v , - - 1)-1S is also a s h a r p l y d o u b l y t r a n s i t i v e s u b s e t of G a n d so c o n t a i n s e v e r y v e c t o r o f t h e f o r m (w, - - 1), w ~ = 0. H e n c e S c o n t a i n s e v e r y v e c t o r o f t h e f o r m (v, - - 1)(w, - - 1) (v - - w, 1), v, w ~= 0. As v h a s m o r e t h a n t w o e l e m e n t s this is e n o u g h t o e n s u r e t h a t V _c S. I f g C S g - i S is also a s h a r p l y d o u b l y t r a n s i t i v e s u b s e t of G so t h a t V c g - i S or g V c S. N o w consider R a n d p u t R , = R [ 7 H . W e t h e n g e t ~ VR,. :Finally s u p p o s e (0, h) f i x e s 0 a n d v. (v, - - l) i n t e r c h a n g e s 0 a n d v so t h a t ( v , - 1)(0, h) does also. H e n c e this is a n i n v o l u t i o n so t h a t h e = 1.

T h i s p r o v e s T h e o r e m 1 in t h e case t h a t p is odd.

N o w s u p p o s e t h a t p --~ 2. I f v =~ 0, (v, 1) i n t e r c h a n g e s 0 a n d v. I f h(v) ~- v, (v, h) also i n t e r c h a n g e s 0 a n d v a n d so is a n i n v o l u t i o n . H e n c e h 2 - - - 1 a u d we d e d u c e t h a t t h e s t a b i l i z e r o f a n y t w o p o i n t s is a n e l e m e n t a r y a b e l i a n 2-group. N o w refer t o t h e r e m a r k of [1, p. 114]. E i t h e r G is a g r o u p of s e m i l i n e a r t r a n s f o r m a t i o n s o v e r a f i e l d or V h a s o r d e r q2 a n d H is i s o m o i p h i e t o L~(q), S u p p o s e t h a t t h e f i r s t is t r u e . W e m a y t a k e V as t h e a d d i t i v e g r o u p o f t h e f i e l d a n d H is t h e n t h e set o f t r a n s f o r m a t i o n s o f t h e f o r m x ~ a x ~ w h e r e ~ is a f i e l d a u t o m o r p h i s m . T h e t r a n s f o r m a t i o n s w h i c h f i x 0 a n d 1 a r e t h o s e o f t h e f o r m x --> x ~ a n d t h e c o n d i t i o n t h a t t h i s b e a n i n v o l u t i o n is a ~ - - - 1. A f i n i t e f i e l d c o n t a i n s a t m o s t one a u t o - m o r p h i s m w h i c h is a n i n v o l u t i o n , so t h a t t h e s t a b i l i z e r o f a n y t w o p o i n t s h a s o r d e r a t m o s t t w o . S u p p o s e t h a t t h e s t a b i l i z e r o f 0 a n d 1 h a s t w o m e m b e r s 1 a n d x - + x ~.

T h e n G c o n t a i n s t w o m e m b e r s i n t e r c h a n g i n g 0 a n d 1, n a m e l y x - - > x ~- 1 a n d x - ~ x ~ zr 1. T h e l a t t e r is c o n j u g a t e to x - ~ x ~ a n d so t h e o n l y i n v o l u t i o n s of G f i x i n g less t h a n 2 s y m b o l s a r e t h o s e of V. T h e s a m e is c l e a r l y t r u e if t h e s t a b i l i z e r of a n y t w o p o i n t s h a s o r d e r 1. N o w consider t h e o t h e r case w h e n V h a s q2 m e m b e r s a n d H is i s o m o r p h i c to L2(q). I f a is a n i n v o l u t i o n o f H t h e n a f i x e s e x a c t l y q s y m b o l s of V a n d as C ( a ) N H h a s o r d e r q, C(a) h a s o r d e r q2. T h u s a h a s q(q2 _ 1) c o n j u g a t e s . T h e n u m b e r o f c o n j u g a t e s in H is q~ - - 1 so t h a t t h e n u m b e r in G - H is ( q - 1)(q ~ - 1). As G is d o u b l y t r a n s i t i v e e a c h of t h e q 2 1 cosets o f H in G - - H c o n t a i n s t h e s a m e n u m b e r o f t h e s e i n v o l u t i o n s . T h i s n u m b e r is q - - 1. As e a c h s u c h coset c o n t a i n s q i n v o l u t i o n s e a c h c o n t a i n s e x a c t l y one m e m b e r f i x i n g less t h a n 2 s y m b o l s o f V a n d t h i s is a m e m b e r of V. I n a n y case we h a v e s h o w n t h a t if p ~ 2 t h e n t h e o n l y i n v o l u t i o n s of G w h i c h f i x less t h a n 2 s y m b o l s a r e t h o s e of V. L e t S b e a n y s h a r p l y d o u b l y t r a n s i t i v e s u b s e t o f G. T h e n S c o n t a i n s o n e s u c h i n v o l u t i o n f r o m e a c h coset o f H in G. H e n c e V c_ S. As in t h e case t h a t p is o d d t h i s is s u f f i c i e n t to e s t a b l i s h T h e o r e m 1.

T h e p r o o f o f T h e o r e m 1 is n o w c o m p l e t e .

P R O J E C T I V E ] ? L A N E S W I T H A C O N D I T I O N O N ] ? I % O J E C T I V I T I E S 145

References

1. ASC~BACHEIr M., Doubly transitive groups in which the stabilizer of two points is abelian.

J . Algebra 18 (1971), 114--136.

2. HUPPERT, B., Endliche Gruppen I , Springer-Verlag, 1968.

3. LOI~IMEIr :P., A note on doubly transitive groups. J. Austral..:Vlath. Soc. 6 (1966), 449--451.

4. TITS, J., Ggn6ralisations des groupes projectifs I, I I , H I , IV. Acad. Roy. Belg. Bull. C1. Sci.

(5) 35 (1949), 197--208, 224--233, 568--589, 756--773.

5. WALTEIL J. H., F i n i t e groups with abelian Sylow 2-groups of order 8. Invent. Math. 2 (1967), 332--376.

J~eceived November 17, 1971 ]~eter Lorimer

D e p a r t m e n t of !Vlathematics U n i v e r s i t y of Auckland P r i v a t e Bag

Auckland New Zealand