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Nonsoluble length of finite groups with commutators of small order

Published online by Cambridge University Press:  13 February 2015

Y. CONTRERAS–ROJAS  and
PAVEL SHUMYATSKY
Y. CONTRERAS–ROJAS
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil. e-mail: Yerkocr@mat.unb.br
PAVEL SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil. e-mail: pavel@unb.br
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Abstract

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length λp(G) is defined as the minimal number of non-p-soluble quotients in a series of this kind.

We deal with the question whether, for a given prime p and a given proper group variety , there is a bound for the non-p-soluble length λp of finite groups whose Sylow p-subgroups belong to . Let the word w be a multilinear commutator. In this paper we answer the question in the affirmative in the case where p is odd and the variety is the one of groups satisfying the law we ≡ 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 3 , May 2015 , pp. 487 - 492
Copyright
Copyright © Cambridge Philosophical Society 2015 

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