Stable Extensions of Complete Groups

TL;DR

This paper classifies finite stable groups formed as central extensions of centerless groups, focusing on those with nontrivial centers and specific extension properties.

Contribution

It provides a comprehensive classification of finite stable groups as central extensions of centerless groups, including those with nilpotency class two and trivial outer action.

Findings

Classified all finite stable groups as central extensions of centerless groups.

Identified stable groups with nontrivial centers arising from specific extensions.

Analyzed extensions with nilpotency class two and trivial outer action.

Abstract

A group is said to be stable if it is isomorphic to its automorphism group. We investigate how we can extend centerless groups to construct finite stable groups with nontrivial centers. To this end, we classify all finite stable groups arising as central extensions of centerless groups. Furthermore, all finite stable groups arising as extensions of centerless groups by groups of nilpotency class two with trivial induced outer action on the kernel are classified.