On Insoluble Transitive Subgroups in the Holomorph of a Finite Soluble Group

TL;DR

This paper classifies irreducible pairs of groups where an insoluble transitive subgroup exists in the holomorph of a finite soluble group, revealing specific structural properties and composition factors.

Contribution

It provides a complete classification of irreducible solutions, identifying the structure of insoluble transitive subgroups in holomorphs of finite soluble groups.

Findings

Every non-abelian composition factor of G is isomorphic to the simple group of order 168.

Every maximal normal subgroup of N has index 2.

The classification covers all irreducible solutions to the problem.

Abstract

A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph $\mathrm{Hol(N)}$ 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 $\mathrm{Hol}(N)$ with soluble point stabilisers. We call such a pair $(G,N)$ irreducible if we cannot pass to proper non-trivial quotients $\overline{G}$, $\overline{N}$ of $G$, $N$ so that $\overline{G}$ becomes a subgroup of $\mathrm{Hol}(\overline{N})$. 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 $2$.