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arXiv · 1907.02185

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

Marius Tărnăuceanu
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Abstract

Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $σ_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $σ_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$. This solves an open problem posed in \cite{8}.

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Marius Tărnăuceanu. 2019-08-11. A criterion for solvability of a finite group by the sum of subgroup orders. https://arxiv.org/abs/1907.02185

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