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Quasi - prime subgroup normalizer well-established group of the exponent of the research

Author: ZhuXiaoXing
Tutor: GuoWenBin
School: Yangzhou University
Course: Basic mathematics
Keywords: Normalizer Quasi - prime subgroup Prime power Sylow Subgroup Group G Finite group Hall subgroup Dedekind Asian nilpotent groups Group class
CLC: O152.1
Type: Master's thesis
Year: 2002
Downloads: 37
Quote: 0
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Abstract


In the discussion of the structure and nature of the finite group G , we often by means of its Subgroups nature . As we all know, Sylow Subgroup normalizer of a finite group plays an extremely important role in the study of a finite group G X represents a group of classes , we scirocco Subgroups normalizer includes the class of the group consisting of the group in x Nx to represent . In 1986 , Bianchi and of Mauri well Hauck First of all nilpotent groups of class N , N ~ N ( ? ) N established . That is, if the group G of all Sylow Subgroup normalizer is nilpotent , then G itself is nilpotent [ 1 ] . In In 1988 , Fedri Serens in [ 2 ] pointed out that for all the ultra- solvable V may not have N ~ V ( ? ) V . Sylow Subgroup normalizer internal properties of finite groups are discussed in the above article . On the other hand , people can also discuss the normalizer of Sylow Subgroup finite group of some kind of external nature . 1988 Kondrat'ev [ 3 ] proved : If the normalizer of any Sylow subgroup of the group G is odd in the index in G , then G is 2 - nilpotent . In 1995 , Zhang [ 4 ] proved that if any of the group G normalizer of Sylow Subgroup well-established number exponent , then G is solvable . Subsequently , Chigira proven in [5] for any r ∈ π (G ) , if p ≠ 3 but (| G: N_G (G_r) |, p) = 1, the group G is p- nilpotent . In 1996 , Guo [ 6 ] proved that a group G Sylow subgroup of the normalizer index was odd or a prime power if and only if G is solvable and G = KH , where K and H are Hall- subgroup of the group G , K is normal in G, a 2'-Hall subgroup nilpotent subgroups H is 2 - nilpotent . As a continuation of the research in this area , this article further research subgroup normalizer well-established quasi- prime number exponent of finite groups . The type of group definitions and key concepts and basic results are given in § 1 , we have the prospective prime subgroup normalizer well-established number of exponential finite group called the NP group . In § 2 , ??we will give the the NP group 's structure : 1 ) If the group G is NP group , then G must be sub - nilpotent groups . 2 ) If the group G is NP group , G is a nilpotent group Dedekind group expansion .

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Group theory > Finite group theory
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