On finite $p$-groups with powerful subgroups

TL;DR

This paper studies the structure of finite p-groups where all subgroups of certain indices are powerful, revealing conditions under which these groups are potent and characterizing groups with all maximal subgroups powerful, including a unique counterexample.

Contribution

It characterizes finite p-groups with powerful subgroups, especially those with all maximal subgroups powerful, and identifies a unique counterexample using computational methods.

Findings

Groups with all maximal subgroups powerful have a regular power structure.

A unique counterexample exists: a 3-group of maximal class and order 81.

The study extends to the case p=2 and other generalizations.

Abstract

In this paper we investigate the structure of finite $p$-groups with the property that every subgroup of index $p^i$ is powerful for some $i$. For odd primes $p$, we show that under certain conditions these groups must be potent. Then, motivated by a question of Mann, we investigate in detail the case when all maximal subgroups are powerful. We show that for odd $p$ any finite $p$-group $G$ with all maximal subgroups powerful has a regular power structure - with precisely one exceptional case which is a $3$-group of maximal class and order $81$. To show this counterexample is unique we use a computational approach. We briefly discuss the case $p=2$ and some generalisations.