Finite Groups Whose Norm Quotient Groups Have Cyclic Sylow Subgroups
Received:February 05, 2021  Revised:October 16, 2021
Key Word: norm   Dedekind group   Hamiltonian group   $\pi$-group   structure of finite group   Sylow subgroup
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11661023) and the Service Industry Development Guide of Guizhou Province (Grant No.QianFaGaiFuWu[2018]1181).
 Author Name Affiliation Songliang CHEN School of Mathematics and Big Data, Guizhou Education University, Guizhou 550018, P. R. China Yun FAN School of Mathematics and Statistics, Central China Normal University, Hubei 430079, P. R. China
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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.