The Genus of a Nilpotent R-Powered Group

Abstract

In [14], Mislin define the genus G(N) of a finitely generated nilpotent group N to be the set of isomorphism classes of finitely generated nilpotent groups M such that the localizations Mp and Np are isomorphic at every prime p. In [6], Hilton and Mislin define an abelian group structure on the genus set G(N) of a finitely generated nilpotent group N with finite commutator subgroups. Let χ0 be the class of finitely generated groups with finite commutator subgroup. For a χ0-group G, the non cancellation set of G, denoted by χ(G), is the set of isomorphism classes of groups H such that G×Z ∼= H ×Z. Warfield, in [19], proved that, if N is a nilpotent χo-group, then G(N) = χ(N). In [20], the author showed that, for a χo-group G the non- cancellation set χ(G) has a group structure similar to the group structure on the Mislin genus of a nilpotent χo-group. Let R be a binomial ring. The nilpotent R-powered group, first introduced by P. Hall in [5], is a nilpotent group G extended by a binomial ring R. Many results that are found in the theory of nilpotent groups carry over to the class of nilpotent R-powered groups. In particular, Majewicz and Zyman in [13], showed that the P-localization of a nilpotent R-powered group G, for a set of primes P in R can be obtained. We study the genus of a finitely R-generated nilpotent R-powered group. We show that the for any two finitely R-generated nilpotent R-powered groups G and H and some finitely R-powered abelian group A, if G(G × A) = G(H × A) then we have G(G) = G(H)