Groups that have the same holomorph as a finite perfect group

TL;DR

This paper characterizes groups sharing the same holomorph as finite perfect groups, providing complete results for centerless cases and highlighting open questions for groups with non-trivial centers.

Contribution

It offers a comprehensive classification of groups with identical holomorphs to finite perfect groups, especially addressing the centerless case and exploring open problems.

Findings

Complete classification for centerless groups

Identification of open questions for groups with non-trivial centers

Examples illustrating peculiarities of the general case

Abstract

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated by a couple of examples that might be of independent interest.