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- 저자 : Zhencai Shen (검색결과 2 건)
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FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS
Shen, Zhencai,Shi, Wujie,Zhang, Jinshan Korean Mathematical Society 2011 대한수학회보 Vol.48 No.6
In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup $\mathcal{A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0=1$. Define $\mathcal{A}_{i+1}(G)/\mathcal{A}_i(G)=\mathcal{A}(G/\mathcal{A}_i(G))$ for $i{\geq}1$. By $\mathcal{A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal{A}_{\infty}(G)$, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in $\mathcal{A}(G)$, then G' is also nilpotent.
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Finite non-nilpotent generalizations of Hamiltonian groups
Zhencai Shen,Wujie Shi,Jinshan Zhang 대한수학회 2011 대한수학회보 Vol.48 No.6
In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup A(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Set A_0=1. Define A_(i+1)(G)/A_i(G)=A(G/A_i(G)) for i≥1. By A_∞(G) denote the terminal term of the ascending series. It is proved that if G=A_∞(G), then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in A(G), then G' is also nilpotent.
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