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ON THE EXPONENT OF A VERBAL SUBGROUP IN A FINITE GROUP

Part of: Representation theory of groups

Published online by Cambridge University Press:  16 April 2013

PAVEL SHUMYATSKY
PAVEL SHUMYATSKY*
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia – DF, Brazil (email: pavel@mat.unb.br)
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Abstract

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Let $w$ be a multilinear commutator word. We prove that if $e$ is a positive integer and $G$ is a finite group in which any nilpotent subgroup generated by $w$-values has exponent dividing $e$, then the exponent of the corresponding verbal subgroup $w(G)$ is bounded in terms of $e$ and $w$only.

Keywords

finite groupsverbal subgroupscommutators

MSC classification

Secondary: 20D25: Special subgroups (Frattini, Fitting, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 3 , December 2012 , pp. 325 - 332
Copyright
Copyright © 2013 Australian Mathematical Publishing Association Inc. 

References

[1]Feit, W. and Thompson, J., ‘Solvability of groups of odd order’, Pacific J. Math. 13 (1963), 773–1029.Google Scholar
[2]Goldschmidt, D., ‘Weakly embedded 2-local subgroups in finite groups’, J. Algebra 21 (1972), 341–351.CrossRefGoogle Scholar
[3]Gorenstein, D., Finite Groups (Chelsea, New York, 1980).Google Scholar
[4]Hall, P. and Higman, G., ‘On the $p$-length of a $p$-soluble group and reduction theorems for Burnside’s problem’, Proc. Lond. Math. Soc. 6(3) (1956), 1–42.CrossRefGoogle Scholar
[5]Jones, G. A., ‘Varieties and simple groups’, J. Aust. Math. Soc. 17 (1974), 163–173.CrossRefGoogle Scholar
[6]Liebeck, M. W., O’Brien, E. A., Shalev, A. and Tiep, P. H., ‘The Ore conjecture’, J. Eur. Math. Soc. 12(4) (2010), 939–1008.CrossRefGoogle Scholar
[7]Mann, A., ‘The exponents of central factor and commutator groups’, J. Group Theory 10(4) (2007), 435–436.CrossRefGoogle Scholar
[8]Nikolov, N. and Segal, D., ‘On finitely generated profinite groups, I: Strong completeness and uniform bounds’, Ann. of Math. (2) 165 (2007), 171–238.CrossRefGoogle Scholar
[9]Shalev, A., ‘Word maps, conjugacy classes, and a noncommutative Waring-type theorem’, Ann. of Math. (2) 170 (2009), 1383–1416.CrossRefGoogle Scholar
[10]Shumyatsky, P., ‘On groups with commutators of bounded order’, Proc. Amer. Math. Soc. 127(9) (1999), 2583–2586.CrossRefGoogle Scholar
[11]Shumyatsky, P., ‘Verbal subgroups in residually finite groups’, Q. J. Math. 51 (2000), 523–528.CrossRefGoogle Scholar
[12]Shumyatsky, P., ‘On varieties arising from the solution of the Restricted Burnside Problem’, J. Pure Appl. Algebra 171 (2002), 67–74.CrossRefGoogle Scholar
[13]Shumyatsky, P., ‘Commutators in residually finite groups’, Israel J. Math. 182(1) (2011), 149–156.CrossRefGoogle Scholar
[14]Shumyatsky, P., ‘Multilinear commutators in residually finite groups’, Israel J. Math. 189(1) (2012), 207–224.CrossRefGoogle Scholar
[15]Turner-Smith, R. F., ‘Marginal subgroup properties for outer commutator words’, Proc. Lond. Math. Soc. 14 (1964), 321–341.CrossRefGoogle Scholar
[16]Turull, A., ‘Fitting height of groups and of fixed points’, J. Algebra 86 (1984), 555–566.CrossRefGoogle Scholar
[17]Zelmanov, E., ‘The solution of the restricted Burnside problem for groups of odd exponent’, Math. USSR Izv. 36 (1991), 41–60.CrossRefGoogle Scholar
[18]Zelmanov, E., ‘The solution of the restricted Burnside problem for 2-groups’, Math. Sb. 182 (1991), 568–592.Google Scholar