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Finite non-nilpotent generalizations of Hamiltonian groups

Zhencai Shen|Wujie Shi|Jinshan Zhang
대한수학회 (The Korean Mathematical Society)|2011.11|대한수학회보 (Bulletin of the KMS)|제48권 제6호|pp. 1147-1155

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키워드

solvable groupderived subgroupmeta-nilpotent groupnilpotency classfitting length
주제분류수학

초록

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.

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 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.

참고문헌 (27)

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