On insoluble transitive subgroups in the holomorph of a finite soluble group

Abstract

A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph Hol(N) of a finite soluble group N can contain an insoluble regular subgroup. We investigate the more general problem of finding an insoluble transitive subgroup G in Hol(N) with soluble point stabilisers. We call such a pair (G,N) irreducible if we cannot pass to proper non-trivial quotients G‾, N‾ of G, N so that G‾ becomes a subgroup of Hol(N‾). We classify all irreducible solutions (G,N) of this problem, showing in particular that every non-abelian composition factor of G is isomorphic to the simple group of order 168. Moreover, every maximal normal subgroup of N has index 2.