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Semipermutable subgroups and s-semipermutable subgroups in finite groups

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Abstract

Suppose that H is a subgroup of a finite group G. We call H is semipermutable in G if HK = KH for any subgroup K of G such that (∣H∣, ∣K∣) = 1; H is s-semipermutable in G if HGp = GpH, for any Sylow p-subgroup Gp of G such that (∣H∣, p) = 1. These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and gives some thoughts in the research process.

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Acknowledgements

The author is extremely grateful to Professor Zhang Qinhai for his solicitation and patient guidance to the manuscript. Thanks for the help of Dr. Wang Lifang, Dr. Su Ning and Ph.D student Lv Yubo.

This work was supported in part by the project of NSF of China (12071092) and the Science and Technology Program of Guangzhou Municipality, China (201804010088).

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  1. School of Mathematics, Guangdong University of Education, Guangzhou, 510310, China

    Yangming Li

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Translated from Advances in Mathematics (China), 2020, 49(4): 385–400

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Li, Y. Semipermutable subgroups and s-semipermutable subgroups in finite groups. Front. Math. China 17, 23–46 (2022). https://doi.org/10.1007/s11464-022-1002-5

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