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

Bull. Korean Math. Soc. 2020 Vol. 57, No. 6, 1475-1479 https://doi.org/10.4134/BKMS.b200004 Published online September 3, 2020 Printed November 30, 2020

Marius T\u arn\u auceanu ``Al. I. Cuza'' University

Abstract : Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. 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 sol\-va\-ble. This partially solves an open problem posed in \cite{9}.