The Reidemeister spectrum of ZM-groups

Abstract

Given a group $G$ and an automorphism $\phi$ of $G$, two elements $x,y\in G$ are said to be $\phi$-conjugate if $x=gy\phi (g{)}^{-1}$ for some $g\in G$. The number $R(\phi )$ of equivalence classes with respect to this relation is called the Reidemeister number of $\phi$ and the set $\{R(\phi )|\phi \in \mathrm{Aut}(G)\}$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectrum of ZM-groups, extending some results of \cite{13}.

Accepted for publication in Bull. Aust. Math. Soc