The multiple holomorph of a semidirect product of groups having coprime exponents

TL;DR

This paper investigates the structure of the multiple holomorph of groups formed as semidirect products with coprime exponents, introducing a new method to construct elements of odd order in the associated quotient group.

Contribution

It presents a novel approach for constructing elements of odd order in the quotient of the multiple holomorph for specific semidirect product groups.

Findings

The quotient T(G) often has order a power of 2, but exceptions exist.

A new construction method for odd order elements in T(G) when G=A⋉C_d.

Applicable to groups where A has finite exponent coprime to d.

Abstract

Given any group $G$, the multiple holomorph $\mathrm{NHol}(G)$ is the normalizer of the holomorph $\mathrm{Hol}(G) = \rho(G)\rtimes \mathrm{Aut}(G)$ in the group of all permutations of $G$, where $\rho$ denotes the right regular representation. The quotient $T(G) = \mathrm{NHol}(G)/\mathrm{Hol}G)$ has order a power of $2$ in many of the known cases, but there are exceptions. We shall give a new method of constructing elements (of odd order) in $T(G)$ when $G=A\rtimes C_d$, where $A$ is a group of finite exponent coprime to $d$ and $C_d$ is the cyclic group of order $d$.