Doubly transitive permutation groups in which the one-point stabilizer is triply transitive on a set of blocks (original) (raw)

Suppose that G is a doubly transitive permutation group on a finite set Q and that for o! in 52 the stabilizer G, has a set C = {B, ,..., B,} of nontrivial blocks of of imprimitivity in 52-{a}, that is, 1 Bi > 1 and / C i > 1. In two previous papers [5, 61 it was shown that apart from a few known groups, the setwise stabilizer of B, in G, acts faithfully on B, if G," is the alternating or symmetric group, one of the Mathieu groups, or a normal extension of PSL(2,q) in their usual representations. The raises the question: QUESTION. If G," is multiply transitive, is it possible to characterize the groups G for which the setwise stablizer in G, of the block B of 2 does not act faithfully on B ? The only groups I know of in which G, is 2-transitive on z1 and for B in Z, the stablizer of B is not faithful on B, are the following: (i) PSL(n, Q) < G < PrL(n, q), for n > 3 in its natural representation. If B E Z then B u (a> is a line and PSL(n-I, q)-< G," < PTL(n-1, 4). (ii) PSU(3, q) < G < PlYJ(3, q) p ermuting the set of absolute points of the projective plane over a field of q2 elements. If B E 1: then B u {a} is a nonabsolute line, 1 B 1 = q, 1 Z 1 = q2, and G,z has a regular normal subgroup of order q?. (iii) G = A, acting on the 15 points of the projective geometry of dimension 3 over a field of two elements; G, N PSL(2, 7) acts 2-transitively on the set of lines containing 01. (iv) G has a regular normal subgroup. If we drop the assumption that G," is 2-transitive, then we have another family of examples. (v) A group G of Ree type R(q) is 2-transitive on q3 + 1 points. For any two distinct points 01, p, there is a unique nontrivial element g in Go,* which 433