The Reidemeister spectrum of split metacyclic groups

Abstract

Given a group $G$ and an automorphism $φ$ of $G$, two elements $x, y \in G$ are said to be $φ$-conjugate if $x = gy φ(g)^{-1}$ for some $g \in G$. The number of equivalence classes for this relation is the Reidemeister number $R(φ)$ of $φ$. The set $\{R(ψ) \mid ψ\in \mathrm{Aut}(G)\}$ is called the Reidemeister spectrum of $G$. We fully determine the Reidemeister spectrum of split metacyclic groups of the form $C_{n} \rtimes C_{p}$ where $p$ is a prime and the action is non-trivial.