Abstract
For a positive integer n, we denote by \(\pi (n)\) the set of prime divisors of n. For a group G and \(a \in G\), we denote by o(a) the order of the element a. We prove that a finite group G is nilpotent if and only if \(\pi (o(ab))=\pi (o(a)o(b))\) for all a, \(b\in G\) of coprime orders.
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Acknowledgements
J. Lu is supported by the Guangxi Natural Science Foundation Program (2020GXNSFAA238045), the Science and technology project of Guangxi (Guike AD21220114), and the Innovation Project of Guangxi Graduate Education (XJCY2023018). B. Zhang is supported by the National Natural Science Foundation of China (12301022), and the Guangxi Basic Ability Promotion Project for Young and Middle-aged Teachers (2020KY02019).
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Li, B., Lu, J., Pang, L. et al. A criterion for nilpotency in finite groups. Arch. Math. 122, 13–16 (2024). https://doi.org/10.1007/s00013-023-01925-3
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DOI: https://doi.org/10.1007/s00013-023-01925-3