Finite $p$-groups of class two with a large multiple holomorph

TL;DR

This paper studies the structure of the quotient of the multiple holomorph by the holomorph for finite p-groups of class two, revealing new prime factors in its order and embedding arbitrary finite groups.

Contribution

It provides explicit examples of finite p-groups of class two with large multiple holomorphs containing complex subgroup structures and introduces methods to embed any finite group into these holomorphs.

Findings

Existence of p-groups with T(G) containing large GL groups

Prime factors of |T(G)| can include primes other than p(p-1)

Any finite group can be embedded into T(G) for suitable G.

Abstract

Let $G$ be any group. The quotient group $T(G)$ of the multiple holomorph by the holomorph of $G$ has been investigated for various families of groups $G$. In this paper, we shall take $G$ to be a finite $p$-group of class two for any odd prime $p$, in which case $T(G)$ may be studied using certain bilinear forms. For any $n\geq 4$, we exhibit examples of $G$ of order $p^{n+{n\choose 2}}$ such that $T(G)$ contains a subgroup isomorphic to \begin{equation*} \operatorname{GL}_n(\mathbb{F}_p) \times \operatorname{GL}_{\binom{n}{2}-n}(\mathbb{F}_p).\end{equation*} For finite $p$-groups $G$, the prime factors of the order of $T(G)$ which are known so far all came from $p(p-1)$. Our examples show that the order of $T(G)$ can have other prime factors as well. In fact, we can embed any finite group into $T(G)$ for a suitable choice of $G$.