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

Abstract

We consider the quotient group $T(G)$ of the multiple holomorph by the holomorph of a finite $p$-group $G$ of class two for an odd prime $p$. By work of the first-named author, we know that $T(G)$ contains a cyclic subgroup of order $p^{r-1}(p-1)$, where $p^r$ is the exponent of the quotient of $G$ by its center. In this paper, we shall exhibit examples of $G$ (with $r=1$) such that $T(G)$ has order exactly $p-1$, which is as small as possible.

Fixed some typos and a small mistake in the introduction

References in corpus (2)

• The multiple holomorph of a finitely generated abelian group
• Finite $p$-groups of class two with a large multiple holomorph

Cited by in corpus (1)

• Transfinite hypercentral iterated wreath product of integral domains