Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2020; 57(6): 1475-1479

Online first article September 3, 2020      Printed November 30, 2020

https://doi.org/10.4134/BKMS.b200004

Copyright © The Korean Mathematical Society.

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

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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