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2023•International Journal of Algebra and ComputationRequires access

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

Osnel Broche, Diego García-Lucas, Ángel del Rı́o

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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.

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What this paper is about

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.

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Available 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.

Key concepts: Cyclic group, Abelian group, Mathematics, Prime power, Prime (order theory), Isomorphism (crystallography), Generator (circuit theory), Generating set of a group

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