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

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 ${\kappa }_1(G)⩽p$ is also determined.