On locally finite groups with a four-subgroup whose centralizer is small

被引：0
作者：
Enio Lima
Pavel Shumyatsky
机构：
[1] University of Brasilia,Department of Mathematics
来源：
Monatshefte für Mathematik | 2013年 / 172卷
关键词：
Locally finite groups; Centralizers; 20F50; 20E2;
D O I：
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学科分类号：
摘要：
Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} be a locally finite group which contains a non-cyclic subgroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V$$\end{document} of order four such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{G}\left( V\right) $$\end{document} is finite and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{G}\left( \phi \right)$$\end{document} has finite exponent for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in V$$\end{document}. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[G,\phi ]^{\prime }$$\end{document} has finite exponent. This enables us to deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} has a normal series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le G_1\le G_2\le G_3\le G$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G/G_2$$\end{document} have finite exponents while \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2/G_1$$\end{document} is abelian. Moreover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_3$$\end{document} is hyperabelian and has finite index in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}.
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页码：77 / 84
页数：7
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共 29 条
[1]  
Feit W(1963)Solvability of groups of odd order Pacific J. Math. 13 773-1029
[2]  
Thompson JG(1976)On orders of finite groups and centralizers of p-elements Osaka J. Math. 13 483-489
[3]  
Fong P(1980)Periodic groups in which the centralizer of an involution has bounded order J. Algebr. 64 285-291
[4]  
Hartley B(1981)Finite soluble groups in which the centralizer of an element of prime order is small Arch. Math. (Basel) 36 211-213
[5]  
Meixner T(1982)Periodic locally soluble groups containing an element of prime order with Cernikov centralizer Q. J. Math. Oxford 33 309-323
[6]  
Hartley B(1992)A general Brauer-Fowler theorem and centralizers in locally finite groups Pacific J. Math. 152 101-117
[7]  
Meixner T(1970)Strong finiteness conditions in locally finite groups Math. Z. 117 309-324
[8]  
Hartley B(1990)Groups and Lie rings admitting an almost regular automorphism of prime order Math. USSR Sbornik 71 51-63
[9]  
Hartley B(2007)Large characteristic subgroups satisfying multilinear commutator identities J. Lond. Math. Soc. 75 635-646
[10]  
Kegel OH(2007)The exponents of central factor and commutator groups J. Group Theory 10 435-436
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