A criterion for solvability of a finite group by the sum of subgroup orders

Abstract

Let $G$ be a finite group and ${\sigma }_1(G)=\frac{1}{|G|}{\sum }_{H\le G}|H|$. In this note, we prove that if ${\sigma }_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have ${\sigma }_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$. This solves an open problem posed in \cite{8}.

arXiv commentAn error occurred in the proof of the main result. A new proof will be given as soon as possible