p-groups having a unique proper non-trivial characteristic subgroup

TL;DR

This paper classifies finite p-groups with exactly three characteristic subgroups, exploring their structure based on exponent and generator count, and relates their automorphism groups to maximal linear groups.

Contribution

It provides new classification theorems for groups with three characteristic subgroups, especially for 3- and 4-generator groups, and examines their automorphism groups in relation to linear groups.

Findings

Classification of 3- and 4-generator groups with three characteristic subgroups.

Existence results for r-generator groups with exponent p^2.

Automorphism groups linked to maximal linear groups in Aschbacher's scheme.

Abstract

We consider the structure of finite $p$-groups $G$ having precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. The structure of $G$ varies markedly depending on whether $G$ has exponent $p$ or $p^2$, and, in both cases, the study of such groups raises deep problems in representation theory. We present classification theorems for 3- and 4-generator groups, and we also study the existence of such $r$-generator groups with exponent $p^2$ for various values of $r$. The automorphism group induced on the Frattini quotient is, in various cases, related to a maximal linear group in Aschbacher's classification scheme.