A classification of the finite two-generated cyclic-by-abelian groups of prime power order

TL;DR

This paper classifies all finite two-generated cyclic-by-abelian groups of prime power order using numerical invariants, enabling their systematic construction, analysis, and isomorphism testing.

Contribution

It introduces a complete classification scheme based on invariants, facilitating computational identification and comparison of these groups.

Findings

Complete list of invariants for classification

Algorithm for constructing all such groups

Method for testing isomorphism between groups

Abstract

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$. Then we describe the set formed by all the possible values of $\inv(G)$. This allows computer implementations for constructing all the finite-two generated cyclic-by-abelian groups of a given prime-power order, computing the invariants of such a group, and to decide whether two such groups are isomorphic.