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Linear groups with almost right Engel elements. (English) Zbl 1457.20029

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

MSC:

20F45 Engel conditions
20G15 Linear algebraic groups over arbitrary fields
20E25 Local properties of groups
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References:

[1] 1.M.De Falco, F.de Giovanni, C.Musella and Ya.Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc.139 (2011), 385-389. · Zbl 1228.20023
[2] 2.W.Feit and J. G.Thompson, Solvability of groups of odd order, Pacific J. Math.13 (1963), 775-1029. · Zbl 0124.26402
[3] 3.M. S.Garascuk, On the theory of generalized nilpotent linear groups, Dokl. Akad. Nauk BSSR.3 (1960), 276-277.
[4] 4.D. M.Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra21 (1972), 341-351. · Zbl 0265.20014
[5] 5.D.Gorenstein, Finite groups (Chelsea Publishing Company, New York, 1980). · Zbl 0185.05701
[6] 6.K. W.Gruenberg, The Engel structure of linear groups, J. Algebra3 (1966), 291-303. · Zbl 0138.26004
[7] 7.O. H.Kegel and B. F. A.Wehrfritz, Locally finite groups (North-Holland, Amsterdam, 1973). · Zbl 0259.20001
[8] 8.E. I.Khukhro and P.Shumyatsky, Almost Engel compact groups, J. Algebra500 (2018), 439-456. · Zbl 1427.20048
[9] 9.L. A.Kurdachenko, J.Otal and I. Ya.Subbotin, On a generalization of Baer theorem, Proc. Amer. Math. Soc.141 (2013), 2597-2602. · Zbl 1281.20035
[10] 10.Yu.Medvedev, On compact Engel groups, Israel J. Math.185 (2003), 147-156. · Zbl 1048.20011
[11] 11.D. J. S.Robinson, A course in the theory of groups, 2nd edn (Springer-Verlag, New York, 1996).
[12] 12.P.Shumyatsky, Almost Engel linear groups, Monatsh. Math.186 (2018), 711-719. · Zbl 1426.20014
[13] 13.J.Tits, Free subgroups in linear groups, J. Algebra20 (1972), 250-270. · Zbl 0236.20032
[14] 14.B. A. F.Wehrfritz, Infinite linear groups (Springer-Verlag, Berlin, 1973). · Zbl 0261.20038
[15] 15.J. S.Wilson and E. I.Zelmanov, Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra81(1) (1992), 103-109. · Zbl 0851.17007
[16] 16.M.Zorn, Nilpotency of finite groups, Bull. Amer. Math. Soc.42 (1936), 485-486. · JFM 62.0088.10
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