On finite $d$-maximal groups

TL;DR

The paper studies finite $d$-maximal groups, proving they are supersolvable and giving a structural framework via maximal $(p,q)$-pairs. It shows non-nilpotent $d$-maximal groups have order divisible by at most two primes and are built from a $p$-group $P$ together with a cyclic rank-$q^t$ automorphism—an approach captured by the notion of maximal $(p,q)$-pairs and actions through characters. Through regularity and detailed rank analyses, it classifies maximal $(p,q)$-pairs of small rank, leading to a full classification of $3$-maximal finite groups and tight bounds on the possible structure (class, order) of such pairs. A key consequence is that the derived length of $d$-maximal groups of odd order is at most $3$, highlighting strong solvability constraints and enabling explicit case analyses for low-rank pairs and small primes.

Abstract

Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up to isomorphism and only partial results have been proven for larger $d$. In this work, we prove that a $d$-maximal group is supersolvable and we give a characterization of $d$-maximality in terms of so-called maximal $(p,q)$-pairs. Moreover, we classify the maximal $(p,q)$-pairs of small rank obtaining, as a consequence, a full classification of the isomorphism classes of $3$-maximal finite groups.

Theorems & Definitions (71)

• Definition 1.1
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.9
• Theorem 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• Lemma 2.4