On Commutation Semigroups of Dihedral Groups

TL;DR

This paper investigates the structure of commutation semigroups in dihedral groups, providing explicit formulas for their orders, which enhances understanding of their algebraic properties.

Contribution

It introduces explicit formulas for the orders of right and left commutation semigroups in dihedral groups, a novel contribution to algebraic semigroup theory.

Findings

Explicit formulas for orders of P(G) and L(G) in dihedral groups

Characterization of commutation semigroups in dihedral groups

Enhanced understanding of algebraic structure of dihedral groups

Abstract

For G a group and g in G, we define mappings pg(G) and lg(G) from G into G by pg(x)=[x,g] and lg(x)=[g,x]. We let P(G) and L(G) denote the subsemigroups of the set of all mappings from G to G generated by {pg: g in G} and {lg: g in G}, respectively. P(G) and L(G) are called the right and left commutation semigroup of G, respectively. In this paper we will give explicit formulas for the orders of both P(G) and L(G) where G is a dihedral group.