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Transactions of the American Mathematical Society

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Bounds on the nilpotency class of certain semidirect products


Author: Larry Morley
Journal: Trans. Amer. Math. Soc. 159 (1971), 381-390
MSC: Primary 20.52
DOI: https://doi.org/10.1090/S0002-9947-1971-0284512-6
MathSciNet review: 0284512
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Abstract: Gilbert Baumslag has shown that the standard wreath product of $ A$ by $ B$ is nilpotent if and only if $ A$ and $ B$ are $ p$-groups for the same prime $ p, A$ is nilpotent of bounded exponent and $ B$ is finite. L. Kaloujnine and Marc Krasner have shown that the standard (unrestricted) wreath product of $ A$ by $ B$ contains an isomorphic copy of every group $ G$ which is an extension of $ A$ by $ B$. Thus it follows that any extension subject to the above condition on $ A$ and $ B$ is nilpotent. In this paper, the author gives an explicit characterization of the terms of the lower central series of a semidirect product $ W$ of an abelian group by an arbitrary group. He then establishes a formula for an upper bound on the nilpotency class of $ W$ when $ W$ is a semidirect product of an abelian $ p$-group $ X$ of bounded exponent by a finite $ p$-group $ B$. This new bound is given in terms of the exponent of $ X$ and the cycle structure of the factor groups of the lower central series of $ B$.


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0284512-6
Keywords: Nilpotency class, lower central series, semidirect products, $ p$-groups, wreath products
Article copyright: © Copyright 1971 American Mathematical Society