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

Abstract

Let $G$ be a finite group and ${\sigma }_1(G)=\frac{1}{|G|}{\sum }_{H\le G}|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if ${\sigma }_1(G)<\frac{117}{20}$, then $G$ is solvable. This partially solves an open problem posed in \cite{9}.