Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups |
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Authors: | Email author" target="_blank">Guo?XiuyunEmail author K?P?Shum |
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Institution: | (1) Department of Mathematics, Shanxi University, 030006 Taiyuan, shanxi, PR China;(2) Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China (SAR) |
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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.
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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Keywords: | |
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