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Linear Groups with Almost Right Engel Elements

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Other groups of matrices

Published online by Cambridge University Press:  30 January 2019

Pavel Shumyatsky
Pavel Shumyatsky*
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil ([email protected])
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Abstract

Let G be a linear group such that for every g ∈ G there is a finite set ${\cal R}(g)$ with the property that for every x ∈ G all sufficiently long commutators [g, x, x, …, x] belong to ${\cal R}(g)$. We prove that G is finite-by-hypercentral.

Keywords

linear groupsEngel conditionlocally nilpotent groups

MSC classification

Primary: 20E99: None of the above, but in this section
Secondary: 20F45: Engel conditions 20H20: Other matrix groups over fields
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 789 - 797
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1.De Falco, M., de Giovanni, F., Musella, C. and Sysak, Ya., On the upper central series of infinite groups, Proc. Amer. Math. Soc. 139 (2011), 385–389.Google Scholar
2.Feit, W. and Thompson, J. G., Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029.Google Scholar
3.Garascuk, M. S., On the theory of generalized nilpotent linear groups, Dokl. Akad. Nauk BSSR. 3 (1960), 276–277.Google Scholar
4.Goldschmidt, D. M., Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972), 341–351.Google Scholar
5.Gorenstein, D., Finite groups (Chelsea Publishing Company, New York, 1980).Google Scholar
6.Gruenberg, K. W., The Engel structure of linear groups, J. Algebra 3 (1966), 291–303.Google Scholar
7.Kegel, O. H. and Wehrfritz, B. F. A., Locally finite groups (North-Holland, Amsterdam, 1973).Google Scholar
8.Khukhro, E. I. and Shumyatsky, P., Almost Engel compact groups, J. Algebra 500 (2018), 439–456.Google Scholar
9.Kurdachenko, L. A., Otal, J. and Subbotin, I. Ya., On a generalization of Baer theorem, Proc. Amer. Math. Soc. 141 (2013), 2597–2602.Google Scholar
10.Medvedev, Yu., On compact Engel groups, Israel J. Math. 185 (2003), 147–156.Google Scholar
11.Robinson, D. J. S., A course in the theory of groups, 2nd edn (Springer-Verlag, New York, 1996).Google Scholar
12.Shumyatsky, P., Almost Engel linear groups, Monatsh. Math. 186 (2018), 711–719.Google Scholar
13.Tits, J., Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.Google Scholar
14.Wehrfritz, B. A. F., Infinite linear groups (Springer-Verlag, Berlin, 1973).Google Scholar
15.Wilson, J. S. and Zelmanov, E. I., Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra 81(1) (1992), 103–109.Google Scholar
16.Zorn, M., Nilpotency of finite groups, Bull. Amer. Math. Soc. 42 (1936), 485–486.Google Scholar