Orbits classifying extensions of prime power order groups

TL;DR

This paper analyzes the classification of group extensions of prime power order groups using cohomology orbits, providing explicit counts and sizes for specific cases involving small abelian p-groups.

Contribution

It offers a detailed study of orbits in the second cohomology group for extensions of abelian p-groups by cyclic groups, including explicit enumeration for groups generated by up to three elements.

Findings

Computed the number of orbits for specific group extensions.

Determined sizes of these orbits in cases with small abelian p-groups.

Provided a framework for classifying extensions via cohomology orbits.

Abstract

The strong isomorphism classes of extensions of finite groups are parametrized by orbits of a prescribed action on the second cohomology group. We study these orbits in the case of extensions of a finite abelian $p$-group by a cyclic factor of order $p$. As an application, we compute number and sizes of these orbits when the initial $p$-group is generated by at most $3$ elements.