Detecting solvability, supersolvability and CLT properties via an invariant based on conjugacy classes of subgroups

Abstract

For a finite group $G$, denote by $k'(G)$ and $L(G)$ the number of conjugacy classes of subgroups and the subgroup lattice of $G$, respectively. Let $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)$ be the minimum value of $d'(S)$, when $S$ runs through all sections of $G$. In this paper we deduce some criteria on the nature of $G$. We show that if $d^*(G)>\frac{9}{59}$, then $G$ is solvable, while if $d^*(G)>\frac{1}{2}$, then $G$ is a supersolvable group. The last criterion is also valid when replacing ``supersolvable" with ``CLT".