Two Families of Flag-transitive Affine Planes

Two families of flag-transitive nondesarguesian affine planes of odd order are defined, and isomorphisms among the various planes are studied. Thir ty years ago Wagner [5] proved the impor tan t result that every finite flag-transitive affine plane is a t ranslat ion plane. However , since that t ime relatively few classes of such planes have been found. The purpose of this note is to c o m m e n t on two construct ions of flag-transitive affine planes due to Suetake [4]. These construct ions will be generalized slightly and will be shown to produce fairly large numbers of pairwise non i somorph ic planes. Mos t of them are new (compare [1], [4] and the references therein), but those that are 2-dimensional over their kernels were known previously [1], [2]. Let F = GF(q2"), L = GF(q") and K --GF(q), where n > 1 and q = pe is a power of an odd pr ime p. Let a ~ Gal(L/K); we will assume (unless stated otherwise) that o # 1. Th roughou t this note we will identify each such a u t o m o r p h i s m a with a power of q inducing it. Assume that n is odd. Let b ~ F be such that b= b (where bar denotes the involutory a u t o m o r p h i s m of F), so that b 2 -b 6 ~ L. Let s ~ F* have order (q" + 1)(q 1). Write h(x) = x + bx ~ for x ~ F, and consider the set 5~b,¢ = {sih(L)[O <. i <~ q"} of subspaces of F, where F is always viewed as a K-space. This is very slightly more general than Suetake 's construct ion (he assumes that a = q and q" 3 (mod 4)). Fo r e ~ F let fi denote the linear t ransformat ion z -~ ez f rom F to itself. T H E O R E M 1. (i) 5Pb,~ iS a spread, and defines a nondesarguesian translation plane Hb,~. (ii) Hb, a admits a flag-transitive group inducing a cyclic group on the line at infinity. (iii) The number of pairwise nonisomorphic translation planes arising in this *Research supported in part by NSF grant DMS 87-01794 and NSA grant MDA 904-88-H2040. Geometriae Dedicata 41: 191-200, 1992. O 1992 Kluwer Academic Publishers. Printed in the Netherlands. 192 W I L L I A M M. K A N T O R manner is at least ~(n/n* 1)(q"* -1)/2en*, where n/n* is the smallest prime fac tor o f n. Proof. (i, ii) F i r s t of all, h is inject ive: if x + bx ~ -0 wi th x e L t hen a l so x + b-x ~ = O, so t h a t x -O. Thus , ~b,~ cons i s t s of n d i m e n s i o n a l Kspaces . C lea r ly , ~ i n d u c e s a cycle t r ans i t i ve o n ~ , ~ , so t h a t (ii) h o l d s if (i) is p r e s u p p o s e d . A s s u m e t h a t h(x) = s~h(y) wi th x, y e L* a n d 0 < i <~ qn. T h e n (x + bx¢)(x + bx ~) = sisi(y + by~)(y + bye), so t h a t x 2 b2(x¢) 2 = k [ y 2 b2(y~) 2] w i th k = s i ~ e K. T h e n x 2 ky 2 = b2(x 2 ky2y . H o w e v e r , (x 2 ky2) ~1 is the s q u a r e of an elem e n t of L w h e r e a s b 2 is not . C o n s e q u e n t l y , x 2 ky 2 = 0, so t h a t x = my w h e r e m e L a n d m 2 = k. S ince n is odd , so is (qn 1)/(q 1), a n d hence m e K. T h e n m(y + by ~) = my + b(my) ~ = si(y + bye), so t h a t m = s i, (sl) q-1 = 1, a n d hence (q~ + 1)(q 1)[i(q 1), wh e re a s 0 < i ~< qn. Th is c o n t r a d i c t i o n p r o v e s (i). Before p r o v i n g (iii) we will need a n i s o m o r p h i s m cr i t e r ion . Le t H(o-) = { z ' * e l ~ z ~° for z e F I e e L * , ~ p e A u t F } , so t h a t H(a ) i nduces a g r o u p of p e r m u t a t i o n s of bL* = {deF* ] d = d } . N o t e t ha t IH(a)[ = 2en(q ~ 1)/(q" 1, a 1). L E M M A 1. (I) IIb, ~ ~ H b l ~ l . (II) I f IIb, ~ _--__ IIc, ~ then z = a+_l. (III) rib, ~ is nondesarguesian. (IV) YIb,~ = Hc,~ i f and only i f b and c are in the same H(a)-orbit. Proof. The t r a n s f o r m a t i o n z ~ b l z sends x + bx ~ to x ~ + b-~(x") ~-~, so (I) ho lds . If b and c are in the s a m e H ( a ) o r b i t t hen c = a l ~ b ~ for s o m e e e L a n d ~o e A u t F. T h e n (ay e) + c(ey*) ~ = e ( y + bye) ~, so t h a t the t r a n s f o r m a t i o n z -~ ez ~ i nduces an i s o m o r p h i s m lib, ~ ~ I I . . . . w h i c h p r o v e s p a r t of (IV). W e will p r o v e (II), ( I I I) a n d the h a r d e r p a r t of (IV) s i m u l t a n e o u s l y . N o t e t h a t S~,~ a n d II~,~ a re a l so m e a n i n g f u l w h e n z = 1 in wh ich case rI~,~ is d e s a r g u e s i a n (since t h e n x + cx ~ = (1 + c)x). W i t h this in m i n d , a s s u m e t h a t rib, ~ ~ H .... w h e r e we t e m p o r a r i l y a l l ow the pos s ib i l i t y t h a t z = 1. N o t e t h a t K is c o n t a i n e d in the ke rne l of b o t h of these p lanes . (Recal l t ha t the kernel of IIb, ~ is the field cons i s t i ng of al l e n d o m o r p h i s m s of the a b e l i a n g r o u p F t h a t m a p each l ine t h r o u g h 0 i n to itself.) I t fo l lows t h a t the re is a K s e m i l i n e a r t r a n s f o r m a t i o n 9: F ~ F s en d in g Hb,~ to II~,~ a n d hence s end ing ~,~ to ~c,~. Le t So E ( s ) have p r i m e o r d e r a n d gene ra t e F * (s o exis ts by [6]). T h e n a T W O F A M I L I E S O F F L A G T R A N S I T I V E A F F I N E P L A N E S 193 Sylow ]Sol-subgroup of FL(F) is cyclic. Clearly, g conjugates the col l inea t ion g roup of 1-Ib,`" to tha t of IIc, ~. By Sylow's Theorem, we m a y assume tha t g conjugates (~o) to itself. Then g has the form z ~ a z " with ~ e F * and ~0 ~ Aut F (for all z 6 F). Since Hb,`" ----1-Ib~,~ (by the par t of(IV) a l ready verified) and (p-19 is an i somorph i sm from 17b%G to 17 .... by replac ing g by ( p l g we m a y assume tha t (p = 1. Similarly, by replac ing g by g~ ~ for some i we m a y assume tha t och(L) = h(L) ° = h'(L) , where h'(x) = x + c x ~ for x ~ F. Thus, for each x ~ L * there is some y ~ L * such that x + c x ~ = a ( y + by"); and x ~ y defines a pe rmu ta t i on of L*. Then x c x ~ = x + ~ x ~ = ~ ( y + -by`') = ~ ( y by`'). Wri te fl = ~ + ~ and 6 = ~ ~. Then 2 x = f l y + 6 b y ~ and 2 c x ~ = b y + f l b y ~, so tha t c{ f l y + 6 b y ' } ~ = b y + f l b y L Note tha t r , fib ~ L. Then (1) cfl~y ~ + c(rb)~y "~ = 6 y + f i b y " for all y ~ L. Wri te y = u z here, where u, z ~ L. By two app l ica t ions of (1), cfl~u~z ~ + u°~{,~z + f lbz ~ cruz ~} = ~uz + flbu`'z%