On induced morphisms of Mislin genera

Abstract

Let $N$ be a nilpotent group with torsion subgroup $TN$, and let $\alpha: TN\rightarrow \tilde T$ be a surjective homomorphism such that $\operatorname{ker}\alpha$ is normal in $N$. Then $\alpha$ determines a nilpotent group $\tilde N$ such that $T\tilde N=\tilde T$ and a function $\alpha_*$ from the Mislin genus of $N$ to that of $\tilde N$ if $N$ (and hence $\tilde N$) is finitely generated. The association $\alpha\mapsto\alpha_*$ satisfies the usual functorial conditions. Moreover $[N,N]$ is finite if and only if $[\tilde N,\tilde N]$ is finite and $\alpha_*$ is then a homomorphism of abelian groups. If $\tilde N$ belongs to the special class studied by Casacuberta and Hilton (Comm. in Alg. 19(7) (1991), 2051--2069), then $\alpha_*$ is surjective. The construction $\alpha_*$ thus enables us to prove that the genus of $N$ is non-trivial in many cases in which $N$ itself is not in the special class; and to establish non-cancellation phenomena relating to such groups $N$.