## Tohoku Mathematical Journal 2016 March SECOND SERIES VOL. 68, NO. 1

 Tohoku Math. J. 68 (2016), 139-159

Title A NOTE ON RHODES AND GOTTLIEB-RHODES GROUPS

Author Kyoung Hwan Choi, Jang Hyun Jo and Jae Min Moon

(Received April 25, 2014, revised October 6, 2014)
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.

Mathematics Subject Classification. Primary 55Q05; Secondary 55Q70.

Key words and phrases. Fox homotopy group, Gottlieb group, Gottlieb-Fox group, Gottlieb-Rhodes group, Rhodes group.