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http://dx.doi.org/10.4134/BKMS.2011.48.6.1147

FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS  

Shen, Zhencai (LMAM and School of Mathematical Sciences Peking University)
Shi, Wujie (School of Mathematics and Statics Chongqing University of Arts and Sciences)
Zhang, Jinshan (School of Science Sichuan University of Science and Engineering)
Publication Information
Bulletin of the Korean Mathematical Society / v.48, no.6, 2011 , pp. 1147-1155 More about this Journal
Abstract
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;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
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