On finite permutation groups of rank three

Abstract

The classification of the finite primitive permutation groups of rank 3 was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank 3 permutation group G has a unique non-trivial block system $B$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^B$ is 2-transitive and one can also show that the action induced on each block in $B$ is also 2-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank 3 imprimitive groups by studying the case where the induced action of G on a block in $B$ is of affine type. Our main theorem divides these rank 3 groups into four classes, which are defined in terms of the kernel of the action of G on $B$. In particular, we completely determine the rank 3 semiprimitive groups for which $G^B$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if G is rank 3 semiprimitive and $G^B$ is affine, then G has a regular normal subgroup which is a special p-group for some prime p.