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arXiv · 2102.07029

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

Marius Tărnăuceanu
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Abstract

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_4$ and $A_4$ are obtained. We also show that $\sigma_1(G)<c$ does not imply the supersolvability of $G$ for no constant $c\in(2,\infty)$.

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Marius Tărnăuceanu. 2021-02-13. On the supersolvability of a finite group by the sum of subgroup orders. https://arxiv.org/abs/2102.07029

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