1 Introduction

Let G be a group and $\text {Aut}(G)$ be the group of automorphisms of G. The concept of conjugacy in G can be naturally generalised to the so-called twisted conjugacy: given $\varphi \in \text {Aut}(G)$ , two elements $x,y\in G$ are called $\varphi $ -conjugate if there exists $g\in G$ such that $x=gy\varphi (g)^{-1}$ . We define the Reidemeister number $R(\varphi )$ of $\varphi $ as the number of equivalence classes with respect to this relation. Note that $R(\varphi )\in \mathbb {N}_0\cup \{\infty \}$ . Also, we define the Reidemeister spectrum of G to be $\text {Spec}_R(G)=\{R(\varphi ) \mid \varphi \in \text { Aut}(G)\}$ . If $\text {Spec}_R(G)=\{\infty \}$ , we say that G has the $R_{\infty }$ -property.

Twisted conjugacy arises naturally in Nielsen fixed-point theory (see, for example, [Reference Jiang12]), but there is also a strong algebraic interest in its study, in particular, in the study of the $R_{\infty }$ -property (see, for example, [Reference Dekimpe and Senden5, Reference Fel’shtyn, Luchnikov and Troitsky8, Reference Fel’shtyn and Troitsky9, Reference Springer15]). For many groups, either partial or full information is known about their Reidemeister spectrum. It has either been proven that they have the $R_{\infty }$ -property (such as Baumslag–Solitar groups [Reference Fel’shtyn and Gonçalves7] or Thompson’s group [Reference Bleak, Fel’shtyn and Gonçalves1]) or that they do not (such as certain groups of exponential growth [Reference Gonçalves and Wong10] or free groups of infinite rank [Reference Dekimpe and Gonçalves3]), and for some groups, even the complete Reidemeister spectrum has been determined (such as low-dimensional crystallographic groups [Reference Dekimpe, Kaiser and Tertooy4] or finite abelian groups [Reference Senden14]).

The starting point for our discussion is the paper [Reference Senden13], where the Reidemeister spectrum has been determined for split metacyclic groups of the form $(C_n\times C_{p^m})\rtimes C_p$ where p is prime, m and n are nonnegative integers, and n is coprime with p. In what follows, we will deal with ZM-groups, that is, finite groups with all Sylow subgroups cyclic. By [Reference Huppert11, Ch. IV, Satz 2.11], such a group is of type

$$ \begin{align*} \text{ZM}(m,n,r)=\langle a, b \mid a^m = b^n = 1, \ b^{-1} a b = a^r\rangle, \end{align*} $$

where the triple $(m,n,r)$ satisfies the conditions

$$ \begin{align*} (m,n)=(m,r-1)=1 \quad \mbox{and}\quad r^n \equiv 1 \ (\text{mod } m). \end{align*} $$

Throughout the paper, for a finite number of positive integers $a_1$ , $a_2$ , …, $a_n$ , we denote by $(a_1,a_2,\ldots ,a_n)$ their greatest common divisor. It is clear that $|\text {ZM}(m,n,r)|=mn$ and $Z(\text {ZM}(m,n,r))=\langle b^{o_m(r)}\rangle $ , where

$$ \begin{align*} o_m(r)={\min}\{k\in\mathbb{N}^* \mid r^k\equiv 1 \ (\text{mod } m)\} \end{align*} $$

is the multiplicative order of r modulo m. For simplicity of notation, throughout the paper, we will use d instead of $o_m(r)$ . For every nonnegative integer u, we define

$$ \begin{align*} [u]_r= \begin{cases} 1+r+\cdots+r^{u-1}, &u>0,\\ 0, &u=0.\end{cases} \end{align*} $$

Our main result is stated as follows.

Theorem 1.1. We have

$$ \begin{align*} \mathrm{Spec}_R(\mathrm{ZM}(m,n,r))=\Bigg\{ \frac{1}{m} \sum_{v=0}^{m-1}\sum_{\substack{0\leq u\leq n-1 \\ {n}/{(n,y-1)}\mid u}}\frac{(m,[u]_r)}{o_{{(m,[u]_r)}/{(m,[u]_r,v)}}(r)}\ \bigg| \ 0\leq y<n, y\equiv 1 \pmod d \Bigg\}. \end{align*} $$

The following is a particular case of Theorem 1.1.

Corollary 1.3. If n is prime, then

$$ \begin{align*} \mathrm{Spec}_R(\mathrm{ZM}(m,n,r))=\bigg\{n-1+ \frac{S}{n}\bigg\}, \end{align*} $$

where

$$ \begin{align*} S= \sum_{u=0}^{n-1}\,(m,[u]_r). \end{align*} $$

Example 1.4. For $n=2$ and $r=m-1$ , we have $\text {ZM}(m,n,r)\cong D_{2m}$ , the dihedral group of order $2m$ . Then, $S=m+1$ and so the formula in Corollary 1.3 becomes

$$ \begin{align*} \text{Spec}_R(D_{2m})=\bigg\{ \frac{m+3}{2}\bigg\} \quad\mbox{for all odd integers } m\geq 3. \end{align*} $$

We remark that a split metacyclic group $(C_n\times C_{p^m})\rtimes C_p$ becomes a ZM-group for $m=0$ and so Theorem 1.1 extends the results obtained in [Reference Senden13] in this particular case.

Our approach uses some basic tools of group action theory, such as Burnside’s lemma, and modular arithmetic.

Burnside’s lemma. Let G be a finite group acting on a finite set X and let

$$ \begin{align*} \mathrm{Fix}(g)=\{x\in X \mid g\circ x=x\} \quad\mbox{for all } g\in G. \end{align*} $$

Then, the number of distinct orbits is

$$ \begin{align*} k=\frac{1}{|G|}\sum_{g\in G}|\mathrm{Fix}(g)|. \end{align*} $$

Most of our notation is standard and will not be repeated here. Basic definitions and results on group theory can be found in [Reference Huppert11].