On some questions related to integrable groups

TL;DR

This paper investigates the classification of finite almost-simple groups and 2-homogeneous subgroups of symmetric groups that are integrable, meaning they are derived subgroups of larger groups, addressing two specific open problems.

Contribution

It classifies almost-simple finite groups and 2-homogeneous subgroups of symmetric groups that are integrable, extending understanding of their structure and integrability conditions.

Findings

Almost-simple finite groups integrable within automorphism groups are classified.

All 2-homogeneous subgroups of symmetric groups that are integrable are characterized.

The classification results solve two previously posed open problems.

Abstract

A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of $G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if $G=H'$ for some $H\leq U$. In this work we focus on two problems posed in [1]. We classify the almost-simple finite groups $G$ that are integrable, which we show to be equivalent to those integrable within $\mathrm{Aut}(S)$, where $S$ is the socle of $G$. We then classify all $2$-homogeneous subgroups of the finite symmetric group $S_n$ that are integrable within $S_n$.