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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.
A CHARACTERIZATION OF SOME PGL(2, q) BY MAXIMUM ELEMENT ORDERS
LI, JINBAO,SHI, WUJIE,YU, DAPENG Korean Mathematical Society 2015 대한수학회보 Vol.52 No.6
In this paper, we characterize some PGL(2, q) by their orders and maximum element orders. We also prove that PSL(2, p) with $p{\geqslant}3$ a prime can be determined by their orders and maximum element orders. Moreover, we show that, in general, if $q=p^n$ with p a prime and n > 1, PGL(2, q) can not be uniquely determined by their orders and maximum element orders. Several known results are generalized.
A CHARACTERIZATION OF SOME PGL(2, q) BY MAXIMUM ELEMENT ORDERS
Jinbao Li,Wujie Shi,Dapeng Yu 대한수학회 2015 대한수학회보 Vol.52 No.6
In this paper, we characterize some PGL(2, q) by their orders and maximum element orders. We also prove that PSL(2, p) with p ≥ 3 a prime can be determined by their orders and maximum element orders. Moreover, we show that, in general, if q = pn with p a prime and n > 1, PGL(2, q) can not be uniquely determined by their orders and maximum element orders. Several known results are generalized.
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.
CONJUGACY SEPARABILITY OF GENERALIZED FREE PRODUCTS OF FINITELY GENERATED NILPOTENT GROUPS
Wei Zhou,김관수,Wujie Shi,C. Y. Tang 대한수학회 2010 대한수학회보 Vol.47 No.6
In this paper, we prove a criterion of conjugacy separability of generalized free products of polycyclic-by-finite groups with a non-cyclic amalgamated subgroup. Applying this criterion,we prove that certain generalized free products of polycyclic-by-finite groups are conjugacy separable.
CONJUGACY SEPARABILITY OF GENERALIZED FREE PRODUCTS OF FINITELY GENERATED NILPOTENT GROUPS
Zhou, Wei,Kim, Goan-Su,Shi, Wujie,Tang, C.Y. Korean Mathematical Society 2010 대한수학회보 Vol.47 No.6
In this paper, we prove a criterion of conjugacy separability of generalized free products of polycyclic-by-finite groups with a non cyclic amalgamated subgroup. Applying this criterion, we prove that certain generalized free products of polycyclic-by-finite groups are conjugacy separable.