有限生成的幂零群的共轭分离性质 |
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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. |
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作者姓名: | 王玉雷 刘合国 张继平 |
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作者单位: | 湖北大学;北京大学数学科学学院 |
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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.
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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 |
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Institution: | The School of
Mathematical Sciences, Peking University, Beijing 100871 |
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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 $G$. |
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Keywords: | nilpotent group conjugacy conjugate separability ■-residual |
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