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

A. Caranti, Cindy Tsang

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.

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

Abstract

We consider the quotient group of the multiple holomorph by the holomorph of a finite -group of class two for an odd prime . By work of the first-named author, we know that contains a cyclic subgroup of order , where is the exponent of the quotient of by its center. In this paper, we shall exhibit examples of (with ) such that has order exactly , which is as small as possible.
Paper Structure (9 sections, 26 theorems, 132 equations)

This paper contains 9 sections, 26 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Multiple holomorph via bilinear forms
  3. A special family of finite $p$-groups of class two
  4. $3$-dimensional case
  5. 4-dimensional case
  6. Proof of the main theorem
  7. A module-theoretic approach
  8. Computation of $S$ and $S'$
  9. The structure of $T(G)$

Key Result

Theorem 1.1

The following holds.

Theorems & Definitions (49)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 39 more