Proportion of Simple Subgroups in Finite Groups and Their Applications

This work introduces and investigates the function $\mathcal{V}(G)=\frac{\text{Simp}(G)}{\mathrm{\mid }L(G)\mathrm{\mid }}$, where $\text{Simp}(G)$ denotes the number of simple subgroups and $\mathrm{\mid }L(G)\mathrm{\mid }$ the total number of subgroups of a finite group $G$. The function $\mathcal{V}(G)$, defined on the interval $[0,1]$, represents the proportion of simple subgroups relative to the total number of subgroups. It serves as a tool for analyzing structural patterns in finite groups, particularly in p-groups and other families.