distance transitive graphs and finite simple groups - Semantic Scholar

DISTANCE TRANSITIVE GRAPHS AND FINITE SIMPLE GROUPS CHERYL E. PRAEGER, JAN SAXL, and KAZUHIRO YOKOYAMA [Received 28 May 1985—Revised 26 August 1986]

Introduction This paper represents the first step in the classification of finite primitive distance transitive graphs. In it we reduce the problem to the case where the automorphism group is either almost simple or affine. Let ^ be a simple, connected, undirected graph with vertex set Q. If oc, /? e Q, then d(a, j8) denotes the distance between a and /3 in (§. Let G be some group of automorphisms of c§. Then ^ is G-distance transitive if for any two pairs of vertices \ 2 and arbitrarily large diameter are included in the following list (which we owe to [2,7]). Examples of primitive transitive graphs. Where nothing is said to the contrary, v denotes the valency and d denotes the diameter of each graph listed. 1. Examples with G = SV. Let 2 be the set of v points on which G acts naturally. A.M.S. (1980) subject classification: 05 C 25, 20 B 25. Proc. London Math. Soc. (3) 55 (1987) 1-21.

CHERYL E. PRAEGER, JAN SAXL, AND KAZUHIRO YOKOYAMA

(a) Let Q = I I be the set of d-subsets of 2 where d < \v- Two vertices A and \dJ B are joined when \AOB\=d-l. If v = 2d, the same construction yields an antipodal graph: if we then identify each element of Q with its complement in 2, we obtain a primitive distance transitive graph of diameter [%d\. ; vertices A and B are joined (b) (Odd graphs) Here v = Id + 1 and Q = \dJ when A D B = 0 . The graph has valency d + 1 and diameter d. 2. Examples of linear type At. Let V = V(l + 1, q) be a vector space of dimension / + 1 over the field G¥(q). Let Q be the set of d-dimensional subspaces of V, where d =£ \{l +1). Two vertices A and B are joined when dim(>l HB) = d-1. The graph has valency (qd - l)(q'~d+i - l)ql{q - I) 2 and diameter d. The automorphism group G is the appropriate linear group (of type Ai{q)). 3. Examples of classical type. Let V be a vector space with a non-singular form of one of the types listed in Table 1. In each case Q is the set of totally isotropic subspaces of V of maximal dimension d, and two vertices are joined when 6\m{A fl B) = d - 1. These are primitive distance transitive graphs admitting an automorphism group G of the type listed in Table 1, except that in the case Dd(q) the graph is bipartite: in that case the bipartite half is a primitive distance transitive graph of diameter \^d\ admitting G. TABLE

1. Distance transitive graphs of classical type

Type of G and ^

DimV

Field order

Type of form

BM CM

q q

diq)

2rf + l Id Id

quadratic symplectic quadratic( + )

d+i(q)

2d + 2

A2d(q)

2d + l Id

q q2

D 2D

2

q

q

2

quadratic( — )

hermitian hermitian

4. Wreath examples (Hamming schemes). Let 2 be a set of c points (c*=2), and let Q = 2 d for some d 2s 2. Two vertices A and B are joined if they differ in exactly one coordinate. The graph (called a Hamming graph) has valency v = d(c — 1) and diameter d. It is G-distance transitive with G = Sc wr Sd, and is primitive whenever c 2= 3. If c = 2 then the graph is both antipodal and bipartite. The antipodal quotient—that is, the graph obtained by identifying each vertex with the unique vertex at distance d from it—is a distance transitive graph of valency v and diameter \_\d\. This graph is primitive if d is odd; if d is even, this graph is still bipartite, but the distance-2 graph obtained from it has two isomorphic connected components, each primitive distance transitive of valency and diameter [\d\. Alternatively, the distance-2 graph of the original graph has two isomorphic connected components, each distance transitive of valency j and diameter [2^] • These are primitive if d is odd but are antipodal if d is

DISTANCE TRANSITIVE GRAPHS

3

even: taking antipodal quotients leads to primitive distance transitive graphs of valency ( j and diameter [\d\ if d > 4. Finally, if one of the above graphs has diameter 2 then its complementary graph is also distance transitive. In particular, if d = 2 and c s* 3, the complementary graph of the Hamming graph is distance transitive of valency (c — I) 2 . 5. Affine examples, (a) Let Q be the set of all d x c matrices over GF(q), with 2 =s d *£ c. Two vertices A and B are joined when A — B is a rank-1 matrix. This graph has valency v = (qc — \){qd — l)/(q — 1) and diameter d. It is always primitive and admits G = M. (GL(d, q) x GL(c, q)), where M is Q under addition, M acts on Q by translation, and A{gx, g2) = g\lAg2 for A e Q, gx e GL(d, q), and g2 e GL(c, q). (b) Let Q be the set of c x c skew symmetric matrices over GF(q), where c € {2d, 2d + 1}. Two vertices A and B are joined when A — B has rank 2. This graph has valency and diameter d = [lc\. It is G-distance transitive and primitive with G = M. GL(c, q), where M = (Q, + ) acts on Q by translation, and if A e Q. and g € GL(c, q), then A . g = gTAg (where gT is the transpose of g). (c) Let Q be the set of d x d hermitian matrices over the field GF(#2). Two vertices A and B are joined when A — B has rankl. This graph has valency v = (#2d — \)l{q + 1) and diameter d. It is G-distance transitive and primitive with G = M. GL(c2).

But then Yx(a) must

consist of disjoint cliques like (B), each of size bx (though there could be further edges between cliques). In particular, bx divides kx. On the other hand, ax = \T(fi) n r(y)| = \(B U C) - {p, y}| = 2bx - 2, so kx = 3bx — 1. Hence fei = 1, so /:i = 2, a contradiction. The following proposition will be helpful in several steps of the proof of the Theorem: it may be compared with [1, Theorem 1], and in fact uses this result in the proof. PROPOSITION 1.2. There is no non-abelian simple group T with a normal set ?T of involutions such that the product of any two distinct members of ZT either belongs to 3~ or has order m, where m is a fixed integer.

Proof. Suppose that T, ST, and m satisfy the assumptions of the proposition. Let x,y eZT with o(xy) = m. Since (xy)2 = x{yxy), we see that m is 2 or 4, or is odd. It follows now, by the Baer-Suzuki Theorem [17, Theorem 2.66], that in fact m must be odd, so T is an odd transposition group. Now let a, s, t be distinct elements of 3~ with a e C(s) D C{i). Since a{st) = (st)a, we have either o(ast) = 2m or st e ST. The former is however impossible, since as, te3~. Thus the graph formed on ST by drawing an edge between s and t precisely when [s, t] = 1 is a disjoint union of cliques. By [1, Theorem 1] we see that T is one of PSL^g), PSU3(^), and Sz(g) with q even, q > 2. There is a unique class of involutions in PSLj(^); since the elements of orders q - 1 and q +1 are real, there are products of pairs of elements of ST of these orders, contrary to our assumption. A similar argument using [35] deals with T = Sz(#). Finally, any involution in PSU3(