Finite groups with few conjugate classes of minimal non-abelian subgroups

TL;DR

This paper classifies finite non-abelian groups based on the number of conjugate classes of their minimal non-abelian subgroups, providing structural characterizations for specific cases.

Contribution

It determines the structure of finite groups with exactly one conjugate class of minimal non-abelian subgroups and extends results to p-groups with up to p such classes.

Findings

Groups with one conjugate class of minimal non-abelian subgroups are fully characterized.

For p-groups, the structure is determined when the number of such classes is at most p.

Abstract

Let $G$ be a finite non-abelian group and $\kappa_1(G)$ the number of conjugate classes of minimal non-abelian subgroups of $G$. The structure of $G$ with $\kappa_1(G)=1$ is determined. In the case of $G$ being the $p$-groups, the structure of $G$ with $\kappa_1(G)\leqslant p$ is also determined.