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有限生成的幂零群的共轭分离性质
引用本文:王玉雷,刘合国,张继平.有限生成的幂零群的共轭分离性质[J].数学学报,2008,51(5):841-846.
作者姓名:王玉雷  刘合国  张继平
作者单位:湖北大学;北京大学数学科学学院
摘    要:研究了有限生成的幂零群中元素的共轭分离问题.设ω表示全部素数组成的集合,π是ω的非空真子集,G是有限生成的幂零群,则下述三条等价:(i)如果x和y是G中的任意两个不共轭的元素,则x和y在G的某个有限p-商群中不共轭,其中p∈π;(ii)如果x和y是G中的任意两个不共轭的元素,则x和y在G的某个有限π-商群中不共轭;(iii)G的挠子群T(G)是π-群且G/T(G)是Abel群.同时举例说明:设G是有限生成的无挠幂零群,对于任意素数p,x和y都在G的有限p-商群G/G~p中共轭,但x和y在G中不共轭.

关 键 词:幂零群  共轭  共轭分离  (?)-剩余
收稿时间:2007-4-29
修稿时间:2008-5-4

Conjugate Separability in Finitely Generated Nilpotent Groups
Institution:The School of Mathematical Sciences, Peking University, Beijing 100871
Abstract:In this paper, conjugate separability problem in a finitely generated nilpotent group is researched. Let $G$ be a finitely generated nilpotent group, $\pi$ be a nonempty proper subset of set $\omega$ of all primes, then the following three results are equivalent: (i) If $x$ and $y$ aren't conjugate in $G$, then $x$ and $y$ aren't conjugate in some finite $p$-quotient group of $G$, where $p\in \pi$; (ii) If $x$ and $y$ aren't conjugate in $G$, then $x$ and $y$ aren't conjugate in some finite $\pi$-quotient group of $G$; (iii) The torsion subgroup $T(G)$ of $G$ is a $\pi$-group and $G/T(G)$ is abelian.\ Furthermore, an example is given, i.e, let $G$ be a finitely generated torsion-free nilpotent group, $x$ and $y$ are conjugate in the quotient group $G/G^{p}$ for arbitrary prime $p$, but $x$ and $y$ aren't conjugate in $G$.
Keywords:nilpotent group  conjugacy  conjugate separability  ■-residual
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