A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order

Abstract

We classify the finite 2-generator cyclic-by-abelian groups of prime-power order. We associate to each such group [Formula: see text] a list [Formula: see text] of numerical group invariants which determines the isomorphism type of [Formula: see text]. Then we describe the set formed by all the possible values of [Formula: see text]. This allows us to develop practical algorithms to construct all finite non-Abelian 2-generator cyclic-by-abelian groups of a given prime-power order, to compute the invariants of such a group, and to decide whether two such groups are isomorphic.