Group Structure via Subgroup Counts

TL;DR

This paper explores how subgroup count restrictions, combined with prime divisor information, determine key structural properties like nilpotency and solvability in finite groups.

Contribution

It improves previous subgroup-based criteria for group properties by incorporating prime divisor counts and proves the bounds are sharp.

Findings

Restrictions on subgroup counts can enforce nilpotency, supersolvability, and solvability.

The criteria are sharper than earlier subgroup count results.

Existence of non-nilpotent, non-supersolvable, non-solvable groups at the bounds.

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.