# On the multiple holomorph of groups of squarefree or odd prime power order

## TL;DR

This paper investigates the structure of the quotient group T(G) related to the holomorph of finite groups, showing it is elementary 2-abelian for squarefree order groups and not a 2-group for certain p-groups of small nilpotency class.

## Contribution

It proves T(G) is elementary 2-abelian for all finite groups of squarefree order and identifies cases where T(G) is not a 2-group among certain p-groups.

## Key findings

- T(G) is elementary 2-abelian for all finite squarefree order groups.
- T(G) is not a 2-group for some p-groups with nilpotency class at most p-1.
- Results extend understanding of the multiple holomorph structure for specific group classes.

## Abstract

Let $G$ be a group and write $\mbox{Perm}(G)$ for its symmetric group. Define $\mbox{Hol}(G)$ to be the holomorph of $G$, regarded as a subgroup of $\mbox{Perm}(G)$, and let $\mbox{NHol}(G)$ denote its normalizer. The quotient $T(G) = \mbox{NHol}(G)/\mbox{Hol}(G)$ has been computed for various families of groups $G$, and in most of the known cases, it turns out to be elementary $2$-abelian, except for two groups of order $16$ and some groups of odd prime power order and nilpotency class two. In this paper, we shall show that $T(G)$ is elementary $2$-abelian for all finite groups $G$ of squarefree order, and that $T(G)$ is not a $2$-group for certain finite $p$-groups $G$ of nilpotency class at most $p-1$.