Shumyatsky, Pavel Linear groups with almost right Engel elements. (English) Zbl 1457.20029 Proc. Edinb. Math. Soc., II. Ser. 62, No. 3, 789-797 (2019). 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. Cited in 4 Documents MSC: 20F45 Engel conditions 20G15 Linear algebraic groups over arbitrary fields 20E25 Local properties of groups Keywords:linear groups; Engel condition; locally nilpotent groups PDFBibTeX XMLCite \textit{P. Shumyatsky}, Proc. Edinb. Math. Soc., II. Ser. 62, No. 3, 789--797 (2019; Zbl 1457.20029) Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.