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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Elements with trivial centralizer in wreath products


Authors: Wolfgang P. Kappe and Donald B. Parker
Journal: Trans. Amer. Math. Soc. 150 (1970), 201-212
MSC: Primary 20.52
MathSciNet review: 0266999
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Abstract: Groups with self-centralizing elements have been investigated in recent papers by Kappe, Konvisser and Seksenbaev. In particular, if $ G = A$wr$ B$ is a wreath product some necessary and some sufficient conditions have been given for the existence of self-centralizing elements and for $ G = \left\langle {{S_G}} \right\rangle $, where $ {S_G}$ is the set of self-centralizing elements. In this paper $ {S_G}$ and the set $ {R_G}$ of elements with trivial centralizer are determined both for restricted and unrestricted wreath products. Based on this the size of $ \left\langle {{S_G}} \right\rangle $ and $ \left\langle {{R_G}} \right\rangle $ is found in some cases, in particular if $ A$ and $ B$ are $ p$-groups or if $ B$ is not periodic.


References [Enhancements On Off] (What's this?)

  • [1] Wolfgang P. Kappe, On the anticenter of nilpotent groups, Illinois J. Math. 12 (1968), 603–609. MR 0237645
  • [2] -, Self-centralizing elements in regular $ p$-groups, (to appear).
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  • [4] Peter M. Neumann, On the structure of standard wreath products of groups, Math. Z. 84 (1964), 343–373. MR 0188280
  • [5] K. Seksenbaev, On the anticenter of bundles of groups, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. Nauk 1966 (1966), no. 1, 20–24 (Russian, with Kazakh summary). MR 0202812

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0266999-7
Keywords: Wreath products, self-centralizing element, element with trivial centralizer, anticenter, $ p$-group
Article copyright: © Copyright 1970 American Mathematical Society