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

Abstract

Let $G$ be a finite group and ${\sigma }_1(G)=\frac{1}{|G|}{\sum }_{H\le G}|H|$. 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 $c\in (2,\infty )$.