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 ...
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.
