In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

中文翻译：

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有限群乘积的结果变化

2014 年，Baumslag 和 Wiegold 证明了有限群G是幂零当且仅当○(xy) =○(X)○(是的) 对于每个X,是的∈G和 （○(X),○(是的)) = 1。这导致了许多结果，这些结果用元素阶的素数除数来表征一个群的幂零性（或幂零霍尔子群的存在，或正常霍尔子群的存在）。在这里，我们以新的方式看待这些结果。我们的第一个主要结果断言G是幂零当且仅当○(xy) ⩽○(X)○(是的) 对于每个X,是的∈G具有（○(X),○(是的)) = 1. 作为直接结果，我们恢复了 Baumslag-Wiegold 定理。这个结果的证明是初级的。我们证明了这个结果的一些变体，这些变体取决于有限单群的分类。