On finite $d$-maximal groups

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.