Group Structure via Subgroup Counts

Abstract

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce nilpotency, supersolvability, and solvability of $G$. These criteria improve earlier results that relied solely on the total number of subgroups, and they are sharp in the sense that for each bound there exist non-nilpotent (respectively non-supersolvable, non-solvable) groups attaining the bound.