The non-cancellation groups of certain groups which are split extensions of a finite abelian group by a finite rank free abelian group

Abstract

The groups we consider in this study belong to the class Xo of all finitely generated groups with finite commutator subgroups. We shall eventually narrow down to the groups of the form T)<lw zn for some nE N and some finite abelian group T. For a Xo-group H, we study the non-cancellation set, X(H), which is defined to be the set of all isomorphism classes of groups K such that H x Z ~ K x Z. For Xo-groups H, on X(H) there is an abelian group structure [38], defined in terms of embeddings of K into H, for groups K of which the isomorphism classes belong to X(H). If H is a nilpotent Xo-group, then the group X(H) is the same as the Hilton-Mislin (see [10]) genus group Q(H) of H. A number of calculations of such Hilton-Mislin genus groups can be found in the literature, and in particular there is a very nice calculation in article [11] of Hilton and Scevenels. The main aim of this thesis is to compute non-cancellation (or genus) groups of special types of .Xo-groups such as mentioned above. The groups in question can in fact be considered to be direct products of metacyclic groups, very much as in [11]. We shall make extensive use of the methods developed in [30] and employ computer algebra packages to compute determinants of endomorphisms of finite groups.