In this note, we investigate the influence of pronormal p-subgroups on the structure of finite groups. We not only simplify, but also generalize some main results of Liu et al. (J Algebra Appl 19:2050110, 2020), Shen et al. (Monatshefte Math 175:629–638, 2014) and Yu and Lai (Monatshefte Math 181:745–747, 2016). We also prove that for p-subgroups of finite groups, the concept of pronormal subgroups, weakly normal subgroups and weakly closed subgroups are equivalent.

中文翻译：

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关于有限群的前正规 p 子群的注记

在本说明中，我们研究了前正态 p 子群对有限群结构的影响。我们不仅简化，而且概括了 Liu 等人的一些主要结果。(J Algebra Appl 19:2050110, 2020)，Shen 等。(Monatshefte Math 175:629–638, 2014) 和 Yu and Lai (Monatshefte Math 181:745–747, 2016)。我们还证明，对于有限群的 p-子群，前正态子群、弱正态子群和弱闭子群的概念是等价的。