On the solvability of a finite group by the sum of subgroup orders
Bull. Korean Math. Soc. 2020 Vol. 57, No. 6, 1475-1479
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}.
Keywords : Subgroup orders, solvable groups
MSC numbers : Primary 20D60; Secondary 20D10, 20F16, 20F17
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