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arXiv · 2210.16666

On the average order of a finite group

Mihai-Silviu Lazorec·Marius Tărnăuceanu
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Abstract

Let $o(G)$ be the average order of a finite group $G$. We show that if $o(G)<c$, where $c\in \lbrace \frac{13}{6}, \frac{11}{4}\rbrace$, then $G$ is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the set containing the average orders of all finite groups is not dense in $[a, \infty)$, for all $a\in [0, \frac{13}{6}]$. We also outline some results related to the integer values of the average order. Since group element orders is a popular research topic, we pose some open problems concerning the average order of a finite group throughout the paper.

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Mihai-Silviu Lazorec, Marius Tărnăuceanu. 2022-10-29. On the average order of a finite group. https://arxiv.org/abs/2210.16666

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