| A Remark on З-Permutability of Finite Groups |
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| 作者姓名： | Li Fang WANG Yan Ming WANG |
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| 作者单位： | [1]School of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China [2]Department of Mathematics, Shanxi Teachers University, Linfen 041004, P. R. China [3]Lingnan College and School of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China [4]Department of Mathematics, Nanchang University, Nanchang 330047, P. R. China |
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| 基金项目： | Project supported by NSF of China （10571181） and Advanced Academic Center of ZSU; Acknowledgements The authors would like to thank the referee for his/her helpful comments. |
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| 摘 要： | Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it permutes with every member of З. Recently, using the Classification of Finite Simple Groups, Heliel, Li and Li proved tile following result：
If the cyclic subgroups of prime order or order 4 iif p = 2） of every member of З are З-permutable subgroups in G, then G is supersolvable.
In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.
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| 关 键 词： | 有限群 超可解群 3-可交换子群 代数学 |
| 收稿时间： | 1 January 2006 |
| 修稿时间： | 2006-01-01 |
A Remark on ℨ-Permutability of Finite Groups |
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| Li Fang WANG Yan Ming WANG.A Remark on ℨ-Permutability of Finite Groups[J].Acta Mathematica Sinica,2007,23(11):1985-1990. |
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| Authors: | Li Fang Wang Yan Ming Wang |
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| Institution: | (1) School of Mathematics, Zhongshan University, Guangzhou, 510275, P. R. China;(2) Department of Mathematics, Shanxi Teachers University, Linfen, 041004, P. R. China;(3) Lingnan College and School of Mathematics, Zhongshan University, Guangzhou, 510275, P. R. China;(4) Department of Mathematics, Nanchang University, Nanchang, 330047, P. R. China |
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| Abstract: | Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, ℨ contains exactly
one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to
be ℨ-permutable if it permutes with every member of ℨ. Recently, using the Classification of Finite
Simple Groups, Heliel, Li and Li proved the following result:
If the cyclic subgroups of prime order or order 4 (if p = 2) of every member of ℨ are ℨ-permutable
subgroups in G, then G is supersolvable.
In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.
Project supported by NSF of China (10571181) and Advanced Academic Center of ZSU |
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| Keywords: | ℨ -permutable subgroups supersolvable groups saturated formations |
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