A note on $d$-maximal $p$-groups I

TL;DR

This paper studies $d$-maximal $p$-groups, generalizes existing results to groups with group actions, and provides new proofs and answers to open questions about minimal non-metacyclic $p$-groups.

Contribution

It extends the theory of $d$-maximal $p$-groups to include group actions and addresses an open question on minimal non-metacyclic $p$-groups.

Findings

Generalized results of Kahn and Laffey to $A$-acted groups

Provided alternative short proofs of existing theorems

Answered a question of Berkovich on minimal non-metacyclic $p$-groups

Abstract

A finite $p$-group $G$ is said to be $d$-maximal if $d(H)<d(G)$ for every subgroup $H<G$, where $d(G)$ denotes the minimal number of generators of $G$. A similar definition can be formulated when $G$ is acted on by some group $A$. We generalize results of B. Kahn and T. Laffey to the latter case, and give them in particular alternative short proofs. We answer moreover a question of Y. Berkovich about the minimal non-metacyclic $p$-groups.