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Permutability of minimal subgroups andp-nilpotency of finite groups

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Abstract

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G N of orderp is permutable inN G (P) and whenp = 2 either every cyclic subgroup ofP ∩G N of order 4 is permutable inN G (P) orP is quaternion-free. Some applications of this result are given.

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Authors and Affiliations

  1. Department of Mathematics, Shanxi University, 030006, Taiyuan, shanxi, PR China

    Guo Xiuyun

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China (SAR)

    K. P. Shum

Authors
  1. Guo Xiuyun
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  2. K. P. Shum
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Corresponding author

Correspondence to Guo Xiuyun.

Additional information

The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China.

The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).

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Xiuyun, G., Shum, K.P. Permutability of minimal subgroups andp-nilpotency of finite groups. Isr. J. Math. 136, 145–155 (2003). https://doi.org/10.1007/BF02807195

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  • DOI: https://doi.org/10.1007/BF02807195

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