Frattini子群循环的有限$p$-群中的非交换集和极大Abel子群
On Non-commuting Sets and Maximal Abelian Subgroups in a Finite $p$-Group with a\Cyclic Frattini Subgroup
Received: January 21, 2014  Revised: July 12, 2014
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 Author Name Affiliation WANG Yulei Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China. LIU Heguo Department of Mathematics, Hubei University, Wuhan 430062, China. WU Zuohui Department of Mathematics, Hubei University, Wuhan 430062, China.
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设$G$是一个群, $X$是$G$的一个子集, 若对于任意$x,y\in X$且$x\neq y$, 都有$xy\neq yx$, 则称$X$是$G$的一个非交换集. 进一步, 如果对于$G$中的任意其它非交换子集$Y$, 都有$|X|\geq|Y|$, 那么称$X$是$G$的一个极大非交换集. 文中确定了Frattini子群循环的有限$p$-\!\!群中极大非交换集和极大Abel子群的势.

Let $G$ be a group. A subset $X$ in $G$ is said to be non-commuting if $xy\neq yx$ for any $x,y\in X$ with $x\neq y$. Further, if $|X|\geq|Y|$ for any other non-commuting subset $Y$ in $G$, then $X$ is said to be a maximal non-commuting set. In this paper, the cardinalities of a maximal non-commuting set and a maximal abelian subgroup in a finite $p$-group with a cyclic Frattini subgroup are determined.
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