# Transitive characteristically simple subgroups of finite quasiprimitive permutation groups

## TL;DR

This paper characterizes nonabelian transitive characteristically simple subgroups within finite quasiprimitive permutation groups, revealing their containment in the socle or base group under specific conditions.

## Contribution

It provides a detailed classification of such subgroups, especially when they do not contain nontrivial normal subgroups of the socle.

## Key findings

- Nonabelian transitive characteristically simple subgroups lie in the socle or base group.
- Explicit descriptions of subgroup possibilities are given under certain conditions.
- The results clarify subgroup structure in finite quasiprimitive permutation groups.

## Abstract

The first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian transitive characteristically simple subgroups $H$ of finite quasiprimitive permutation groups $G$. If the socle of $G$, denoted by $\mbox{soc}(G)$, is nonabelian, then $H$ lies in $\mbox{soc}(G)$. An explicit description is given for the possibilities of $H$ under the condition that $H$ does not contain a nontrivial normal subgroup of $\mbox{soc}(G)$.