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.

中文翻译：

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有限群中的幂零准则

对于正整数n，我们用\(\pi (n)\)表示n的素因数集合。对于群G和\(a \in G\)，我们用o ( a ) 表示元素a的阶。我们证明有限群G是幂零的当且仅当\(\pi (o(ab))=\pi (o(a)o(b))\)对于所有a、\(b\in G\)互素阶。