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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An improvement on the upper bound of the nilpotency class of semidirect products of $ p$-groups


Authors: Larry J. Morley and John D. P. Meldrum
Journal: Proc. Amer. Math. Soc. 60 (1976), 53-56
MSC: Primary 20D40
DOI: https://doi.org/10.1090/S0002-9939-1976-0427470-3
MathSciNet review: 0427470
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Abstract: The semidirect product of a group A by a group B is necessarily nilpotent only in the case A and B are p-groups for the same prime p, A is nilpotent of bounded exponent, and B is finite. In an earlier paper Morley has established an upper bound on the class of a nilpotent semidirect product of an abelian p-group of bounded exponent by an arbitrary finite p-group. In this paper this result is improved by considering a direct product decomposition for B and also by extending the results to give a new upper bound on the class in the most general case. The standard wreath product of A by B is a nilpotent semidirect product of relatively large class in the case A and B satisfy the conditions above, and this new bound improves the known results on the class of these wreath products.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0427470-3
Keywords: Nilpotency class, lower central series, semidirect products, direct product decomposition, p-groups, wreath products
Article copyright: © Copyright 1976 American Mathematical Society