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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)
  • Received : 2010.04.07
  • Published : 2011.11.30

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.

Acknowledgement

Supported by : Natural Science Foundation of China, NSF

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