Computing Mislin genera of certain groups with non-abelian torsion radicals

Abstract

In this mini-thesis we present some generalities of non-cancellation and localization and we compute non-cancellation groups. We consider groups belonging to the class X0 of all finitely generated groups that have finite commutator subgroups. For a X0-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. In particular, we prove some basic facts such as that for a group G which is either finite or finitely generated abelian, we have H x Z ~ G x Z only if G"' H. For a finitely generated nilpotent group N , the Mislin genus, Q(N), is defined to be the set of all isomorphism classes of finitely generated nilpotent groups M such that for every prime p, the groups M and N have isomorphic p-localizations. It was shown by Warfield that if N is a nilpotent X0-group, then x(N) = Q(N). Various calculations of such Hilton-Mislin genus groups can be found in the literature, for example, in an article of Hilton and Scevenels. Most of these calculations are for a special subclass of nilpotent X0-groups, in particular, groups with abelian torsion radicals.