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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.

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Keywords: Wreath products, self-centralizing element, element with trivial centralizer, anticenter, $ p$-group
Article copyright: © Copyright 1970 American Mathematical Society