On the supersolvability of a finite group by the sum of subgroup orders

TL;DR

This paper establishes a criterion based on the sum of subgroup orders that guarantees a finite group is supersolvable, providing new characterizations for specific groups and exploring the limits of this criterion.

Contribution

It introduces a new condition involving the average subgroup order that ensures supersolvability and offers novel characterizations of certain well-known groups.

Findings

If $\sigma_1(G)<2+rac{11}{|G|}$, then G is supersolvable.

Provides new characterizations of groups $ ext{Z}_2 imes ext{Z}_4$ and $A_4$.

The bound on $\sigma_1(G)$ cannot be extended to any constant greater than 2 for guaranteeing supersolvability.

Abstract

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_4$ and $A_4$ are obtained. We also show that $\sigma_1(G)<c$ does not imply the supersolvability of $G$ for no constant $c\in(2,\infty)$.