Commutation Semigroups of Finite Metacyclic Groups with Trivial Centre

Abstract

We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented \[G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in\mathbb{Z}^+)\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\,\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \emph{right} and \emph{left commutation semigroups of} $G,$ denoted ${\rm P}(G)$ and $\Lambda (G),$ are the semigroups of mappings generated by $\rho (g):G \to G$ and $\lambda (g):G \to G$ defined by $(x)\rho (g) = [x,g]$ and $(x)\lambda (g) = [g,x],$ where the commutator of $g$ and $h$ is defined as $[g,h] = {g^{ - 1}}{h^{ - 1}}gh.$ This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to $G,$ a metacyclic group with trivial centre. We give a construction of ${\rm P}(G)$ and $\Lambda (G)$ as unions of \emph{containers}, an idea presented in the previous paper on dihedral groups. In the case that $\left\langle a \right\rangle$ is cyclic of order $p$ or ${p^2}$ or its index is prime, we show that both ${\rm P}(G)$ and $\Lambda (G)$ are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for $G(m,n,k)$ with $m$ prime, the condition $\left| {{\rm P}(G)} \right| = \left| {\Lambda (G)} \right|$ is equivalent to ${\rm P}(G) = \Lambda (G).$