On the holomorph of finite semisimple groups

TL;DR

This paper characterizes all finite nonabelian semisimple groups that share the same holomorph, a specific permutation group structure, as a given group, by analyzing their regular subgroups within the symmetric group.

Contribution

It provides a classification of groups with identical holomorphs to a given finite nonabelian semisimple group, expanding understanding of their automorphism and permutation structures.

Findings

Identifies conditions for groups to have the same holomorph as a given group

Describes the structure of regular subgroups in the symmetric group

Characterizes the automorphism groups related to these holomorphs

Abstract

Given a finite nonabelian semisimple group $G$, we describe those groups that have the same holomorph as $G$, that is, those regular subgroups $N\simeq G$ of $S(G)$, the group of permutations on the set $G$, such that $N_{S(G)}(N)=N_{S(G)}(\rho(G))$, where $\rho$ is the right regular representation of $G$.