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

Bull. Korean Math. Soc. Published online September 3, 2020

MARIUS TARNAUCEANU
FACULTY OF MATHEMATICS, AL. I. CUZA UNIVERSITY OF IASI, ROMANIA

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 solvable. This partially solves an open problem posed in \cite{6}.