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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
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil. e-mail: [email protected]
PAVEL SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil. e-mail: [email protected]

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
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Acciarri, C., Fernández-Alcober, G. A. and Shumyatsky, P.A focal subgroup theorem for outer commutator words. J. Group Theory 15 (2012), 397405.Google Scholar
[2]Detomi, E., Morigi, M. and Shumyatsky, P.Bounding the exponent of a verbal subgroup. Ann. Mat. 193 (2014), 14311441.Google Scholar
[3]Feit, W. and Thompson, J. G.Solvability of groups of odd order. Pacific J. Math. 13 (1963), 7731029.Google Scholar
[4]Hall, P. and Higman, G.The p-length of a p-soluble group and reduction theorems for Burnside's problem. Proc. London Math. Soc. (3) 6 (1956), 142.CrossRefGoogle Scholar
[5]Khukhro, E. I. and Shumyatsky, P. Nonsoluble and non-p-soluble length of finite groups. To appear in Israel J. Math.Google Scholar
[6]Khukhro, E. I. and Shumyatsky, P.Words and pronilpotent subgroups in profinite groups. J. Aust. Math. Soc. 97 (2014), 343364.CrossRefGoogle Scholar
[7]Unsolved Problems in Group Theory. The Kourovka Notebook, no. 17, Institute of Mathematics (Novosibirsk, 2010).Google Scholar
[8]Liebeck, M. W., O' Brien, E. A., Shalev, A. and Tiep, P. H.The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 9391008.Google Scholar
[9]Shumyatsky, P.Verbal subgroups in residually finite groups. Quart. J. Math. 51 (2000) 523528.Google Scholar
[10]Shumyatsky, P.Commutators in residually finite groups. Israel J. Math. 182 (2011), 149156.Google Scholar
[11]Shumyatsky, P.On the exponent of a verbal subgroup in a finite group. J. Aust. Math. Soc. 93 (2012), 325332.Google Scholar
[12]Wilson, J.On the structure of compact torsion groups. Monatsh. Math. 96, 5766.Google Scholar
[13]Zelmanov, E. I.On some problems of the theory of groups and Lie algebras. Mat. Sb. 180, 159167; English transl. Math. USSR Sb. 66 (1990), 159–168.Google Scholar
[14]Zelmanov, E. I.A solution of the Restricted Burnside Problem for groups of odd exponent. Izv. Akad. Nauk SSSR Ser. Mat. 54, 4259; English transl. Math. USSR Izvestiya. 36 (1991), 41–60.Google Scholar
[15]Zelmanov, E. I.A solution of the Restricted Burnside Problem for 2-groups. Mat. Sb. 182, 568592; English transl. Math. USSR Sb. 72 (1992), 543–565.Google Scholar
[16]Zelmanov, E. I.On periodic compact groups. Israel J. Math. 77, no. 1–2 (1992), 8395.Google Scholar