Sylow-子群的自同构导子是小的有限群
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摘要
目的研究自同构导子对有限群结构的影响。方法使用单群分类定理及圈积等手段,分可解群、非可解群两种情况分别讨论。结果证明了Sylow子群的自同构导子是小的有限群恰是幂零群。结论某些特殊子群的自同构导子对有限群的结构具有很强的影响,对自同构导子附加合适条件后,可以得到有限群若干信息。
Objective To investigate the influence of Automizer on the structure of finite groups.Methods By the classification of finite simple groups and wreath product,the groups are divided into solvable and non-solvable groups.Results It proves that finite groups with small automizers for their Sylow subgroups are nilpotent groups.Conclusion The automizers of some subgroups have a strong influence on the structure of finite groups.Adding suitable conditions to automizers can lead to certain information of finite groups.
Objective To investigate the influence of Automizer on the structure of finite groups.Methods By the classification of finite simple groups and wreath product,the groups are divided into solvable and non-solvable groups.Results It proves that finite groups with small automizers for their Sylow subgroups are nilpotent groups.Conclusion The automizers of some subgroups have a strong influence on the structure of finite groups.Adding suitable conditions to automizers can lead to certain information of finite groups.
引文
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[2]NOMURA K.Inner subgroups of finite groups[J].Kodai Math J,1978,1(03):354-361.
[3]BECHTELL H,DEACONESCU M,SILBERBERG G H.Finite groups with large automizers for their abelian subgroups[J].Canad Math Bull,1997,43(03):266-270.
[4]DEACONESCU M,WALLS G.Automotives[M].London:Cambridge University Press,2011:228-243.
[5]BIANCHI M G,Mauri A G B,HAUCK P.On finite groups with nilpotent Sylow-normalizers[J].Archiv der Math,1986,47(03):193-197.
[6]HUPPER B,BLACKBURN N.Finite groups III[M].Heidelberg-New-York-Beilin:Springer,1982:12-74.
[7]BALLESTER-BOLINCHES A,SHEMETKOV L A.On normalizers of Sylow subgroups in finite groups[J].Siber Math J,1999,40(01):1-2.