# Skew product groups for monolithic groups

## TL;DR

This paper classifies skew product groups where the subgroup is monolithic and core-free, leading to a comprehensive classification of proper skew morphisms in finite non-abelian simple groups.

## Contribution

It provides a complete classification of monolithic core-free subgroups in skew product groups and applies this to classify skew morphisms of finite non-abelian simple groups.

## Key findings

- Classified all monolithic core-free subgroups in skew product groups.
- Established a classification of proper skew morphisms in finite non-abelian simple groups.
- Enhanced understanding of group factorizations and automorphism generalizations.

## Abstract

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.