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

2019/07/03 by Marius Tărnăuceanu, Tărnăuceanu, Marius
Engineering · Mathematics · #Finite Group Theory Research #Graph theory and applications #graph theory and CDMA systems #math.GR

paper · pdf · doi:10.48550/arxiv.1907.02185

An error occurred in the proof of the main result. A new proof will be given as soon as possible

arxiv created 2019/08/11 · arxiv updated 2019/08/13

Abstract

Let G be a finite group and σ₁(G)=(1)/(|G|)∑_{H≤ G} |H|. In this note, we prove that if σ₁(G)<(117)/(20), then G is solvable. Moreover, we have σ₁(G)=(117)/(20) if and only if G≅ A₅. This solves an open problem posed in \cite8.

Citations

• Finite groups determined by an inequality of the orders of their subgroups II

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