A note on Rhodes and Gottlieb-Rhodes groups

Abstract

The purpose of this paper is to give positive answers to some questions which are related to Fox, Rhodes, Gottlieb-Fox, and Gottlieb-Rhodes groups. Firstly, we show that for a compactly generated Hausdorff based $G$-space $(X,x_0,G)$ with free and properly discontinuous $G$-action, if $(X,x_0,G)$ is homotopically $n$-equivariant, then the $n$-th Gottlieb-Rhodes group $G\sigma_n(X,x_0,G)$ is isomorphic to the $n$-th Gottlieb-Fox group $G\tau_n(X/G,p(x_0))$. Secondly, we prove that every short exact sequence of groups is $n$-Rhodes-Fox realizable for any positive integer $n$. Finally, we present some positive answers to restricted realization problems for Gottlieb-Fox groups and Gottlieb-Rhodes groups.