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A criterion for nilpotency of a finite group by the sum of element orders
Communications in Algebra ( IF 0.7 ) Pub Date : 2020-12-17 , DOI: 10.1080/00927872.2020.1840575
Marius Tărnăuceanu 1
Affiliation  

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $|G|=n$ and $\psi(G)>\frac{13}{21}\,\psi(C_n)$, then $G$ is nilpotent. Moreover, we have $\psi(G)=\frac{13}{21}\,\psi(C_n)$ if and only if $n=6m$ with $(6,m)=1$ and $G\cong S_3\times C_m$. Two interesting consequences of this result are also presented.

中文翻译:

通过元素阶数之和确定有限群的幂零性的判据

用$\psi(G)$表示有限群$G$中的元素阶数之和,让$C_n$表示阶$n$的循环群。在本文中,我们证明如果 $|G|=n$ 且 $\psi(G)>\frac{13}{21}\,\psi(C_n)$,​​则 $G$ 是幂零的。此外,我们有 $\psi(G)=\frac{13}{21}\,\psi(C_n)$ 当且仅当 $n=6m$ 且 $(6,m)=1$ 和 $G\丛 S_3\times C_m$。还介绍了这个结果的两个有趣的结果。
更新日期:2020-12-17
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