The multiple holomorph of centerless groups

TL;DR

This paper investigates the structure of the multiple holomorph of centerless groups, showing that for such groups, a certain quotient has exponent at most 2, with specific results for almost simple, perfect, and complete groups.

Contribution

It establishes bounds on the exponent of the quotient T(G) for centerless groups and computes its order for almost simple groups, advancing understanding of their automorphism-related structures.

Findings

T(G) has exponent at most 2 for most centerless groups.

T(G) has order 2 for all almost simple groups.

T(G) has exponent at most 2 for all centerless perfect or complete groups.

Abstract

Let $G$ be a group. The holomorph $\mathrm{Hol}(G)$ may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of $G$. The multiple holomorph $\mathrm{NHol}(G)$ is in turn defined as the normalizer of the holomorph. Their quotient $T(G) = \mathrm{NHol}(G)/\mathrm{Hol}(G)$ has been computed for various families of groups $G$. In this paper, we consider the case when $G$ is centerless, and we show that $T(G)$ must have exponent at most $2$ unless $G$ satisfies some fairly strong conditions. As applications of our main theorem, we are able to show that $T(G)$ has order $2$ for all almost simple groups $G$, and that $T(G)$ has exponent at most $2$ for all centerless perfect or complete groups $G$.