Affine Distance-transitive Graphs with Sporadic Stabilizer

Abstract

This paper is a contribution to the programme to classify finite distance-transitive graphs and their automorphism groups. We classify pairs ( , G) where is a graph and G is an automorphism group of acting distance-transitively and primitively on the vertex set of , subject to the condition that there is a normal elementary abelian subgroup V in G which acts regularly on the vertex set of and the stabilizer G 0 of a vertex (which is a complement to V in G) has a unique non-abelian composition factor isomorphic to one of the 26 sporadic simple groups. There are exactly 10 examples of , all known for a long time.

Arithmetic results on orbits of linear groups

“…When r = 11, 13, 17, 23 or 25 and (d, q) = (5, 5), (6,4), (8,2), (11,2) or (12,2), the orbit sizes are given by [7,Section 5] and [30,Appendix 2]; there is an orbit size divisible by p in all cases. Now consider the cases where r = 11, 13, 23 or 31 and (d, q) = (6, 3), (7, 3), (11,3) or (15,2). For these, we observe that there is a nonzero vector fixed by a subgroup H of order 11,7,11 or 5 respectively, and H generates L together with any Sylow p-subgroup, a contradiction.…”

“…This leaves the cases where r = 7 or 11 and (d, q) = (3,9), (5,4) or (10,2). For these cases a Magma computation shows that there is an orbit of size divisible by p.…”

“…Proof Suppose L is an exceptional group. By [24] and (8), the possibilities are: L d q G 2 (3) 14 2 G 2 (4) 12 3, 5, 7 2 B 2 (8) 8 5 For (d, q) = (14, 2), (12,3), (12,5), (12,7), there is a subgroup H of L of order 3 5 , 2 9 , 2 10 , 2 8 , respectively, fixing a 1-space, and H generates L with any Sylow p-subgroup, a contradiction. For the remaining case (d, q) = (8, 5), a Magma computation gives an orbit of size divisible by 5.…”

“…(ii) (n, p) = (8,3), and H = AGL 3 (2) = 2 3 : SL 3 (2) or H = AΓL 1 (8) = 2 3 : 7 : 3; (iii) (n, p) = (5,2) and H = D 10 , a dihedral group of order 10.…”

Affine distance-transitive graphs and classical groups

“…The subcase where H 0 /Z(H 0 ) is a sporadic group is covered in [25]; one finds the examples (g)-(i) of Theorem 1.1 related to the Golay codes and Mathieu groups. Alternating groups are considered by Liebeck and Praeger [18] leading to Hamming graphs and the related halved and folded (n + 1)-cubes as well as folded halved (n + 2)-cubes.…”