The Reidemeister spectrum of ZM-groups
Marius Tărnăuceanu
TL;DR
This paper determines the Reidemeister spectrum ${\rm Spec}_R(G)$ for ZM-groups ${\rm ZM}(m,n,r)$, extending prior work on split metacyclic groups. It classifies automorphisms of ${\rm ZM}(m,n,r)$ as $f_{(x_1,x_2,y)}$ and applies Burnside's lemma to count twisted conjugacy classes, deriving an explicit formula for the spectrum. The main result provides ${\rm Spec}_R({\rm ZM}(m,n,r))$ as an exact set given by a double sum over arithmetic data involving $m,n,r$, with corollaries for prime $n$ and dihedral/dicyclic special cases. These findings extend existing results, illustrate the method via concrete instances, and contribute to the broader study of twisted conjugacy in finite groups.
Abstract
Given a group $G$ and an automorphism $\varphi$ of $G$, two elements $x,y\in G$ are said to be $\varphi$-conjugate if $x=gy\varphi(g)^{-1}$ for some $g\in G$. The number $R(\varphi)$ of equivalence classes with respect to this relation is called the Reidemeister number of $\varphi$ and the set $\{R(\varphi)|\varphi\in {\rm Aut}(G)\}$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectrum of ZM-groups, extending some results of \cite{13}.