Bull. Korean Math. Soc. 2011; 48(6): 1147-1155
Printed November 1, 2011
https://doi.org/10.4134/BKMS.2011.48.6.1147
Copyright © The Korean Mathematical Society.
Zhencai Shen, Wujie Shi, and Jinshan Zhang
Peking University, Chongqing University of Arts and Sciences, Sichuan University of Science and Engineering
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.
Keywords: derived subgroup, meta-nilpotent group, solvable group, nilpotency class, fitting length
MSC numbers: 20D10, 20D15, 20D20, 20D30, 20F14, 20F19
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