Norm商群的Sylow子群皆循环的有限群

Finite Groups Whose Norm Quotient Groups Have Cyclic Sylow Subgroups

中文关键词:  norm(范)  戴德金群  哈密顿群  $\phi$-群  有限群的构造  西洛子群

英文关键词:norm  Dedekind group  Hamiltonian group  $\pi$-group  structure of finite group  Sylow subgroup

中文摘要:

      设$G$是有限群, $N(G)$为$G$的norm, 则$N(G)$是$G$的正规化G的每个子群的特征子群. 我们在下列条件之一下,研究了$G$的结构：1) Norm商群$G/N(G)$是循环群；2) Norm商群$G/N(G)$的所有Sylow子群都是循环群,特别地当$G/N(G)$的阶是无平方因子数时.

英文摘要:

      Let $G$ be a finite group and $N(G)$ be its norm. Then $N(G)$ is a characteristic subgroup of $G$ which normalizes every subgroup of $G$. In this paper, we will study the structure of $G$ under one of the following conditions: 1) norm quotient group $G/N(G)$ is cyclic; 2) all Sylow subgroups of $G/N(G)$ are cyclic and in particular if the order of $G/N(G)$ is a square-free number.