The genus of a direct product of certain nilpotent groups with a finite nilpotent group

摘要

Introduction. In [6] Mislin defined the genus (JV) of a finitely generated nilpotent group N as the set of isomorphism classes of finitely generated nilpotent groups M such that for all primes p the p-localizations of M and N are isomorphic, that is Mp Np for all primes p. If the isomorphism class of M belongs to @(N), we say that M e <&(N). Moreover if N e JfQ, that is, N is a finitely generated infinite nilpotent group with a finite commutator subgroup [N,N], then Hilton and Mislin ([4]) showed that (N) has the structure of a finite abelian group. For the reader's convenience, we gather in the first section the definitions and results that we will need.