Commutation Semigroups of Finite Metacyclic Groups with Trivial Centre

TL;DR

This paper investigates the structure of right and left commutation semigroups of finite metacyclic groups with trivial center, providing explicit descriptions and conditions for their equality, extending previous work on dihedral groups.

Contribution

It extends the analysis of commutation semigroups from dihedral groups to a broader class of metacyclic groups with trivial center, introducing the concept of containers for their structure.

Findings

Both semigroups are unions of maximal containers in certain cases.

Explicit element representations and order formulas are provided.

Equality of semigroups is characterized for prime order groups.

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 ${\rm P}(G) = \Lambda (G).$