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On residually finite groups satisfying an Engel type identity. (English) Zbl 07228143

Summary: Let \(n, q\) be positive integers. We show that if \(G\) is a finitely generated residually finite group satisfying the identity \([x,_n y^q] \equiv 1\), then there exists a function \(f(n)\) such that \(G\) has a nilpotent subgroup of finite index of class at most \(f(n)\). We also extend this result to locally graded groups.

MSC:

20F45 Engel conditions
20E26 Residual properties and generalizations; residually finite groups
20F40 Associated Lie structures for groups
20F50 Periodic groups; locally finite groups
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[1] Abdollahi, A.; Traustason, G., On locally finite \(p\)-group satisfying an Engel condition, Proc. Am. Math. Soc., 130, 2827-2836 (2002) · Zbl 1006.20029 · doi:10.1090/S0002-9939-02-06421-3
[2] Bastos, R.; Shumyatsky, P.; Tortora, A.; Tota, M., On groups admitting a word whose values are Engel, Int. J. Algebra Comput., 23, 1, 81-89 (2013) · Zbl 1277.20039 · doi:10.1142/S0218196712500798
[3] Burns, RG; Medvedev, Y., A note on Engel groups and local nilpotence, J. Austral. Math. Soc. Ser. A, 64, 92-100 (1998) · Zbl 0898.20021 · doi:10.1017/S1446788700001324
[4] Dixon, JD; du Sautoy, MPF; Mann, A.; Segal, D., Analytic Pro-p Groups (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0744.20002
[5] Huppert, B.; Blackburn, N., Finite Groups II (1982), Berlin: Springer, Berlin · Zbl 0477.20001
[6] Lazard, M., Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup., 71, 101-190 (1954) · Zbl 0055.25103 · doi:10.24033/asens.1021
[7] Longobardi, P.; Maj, M.; Smith, H., A note on locally graded groups, Rend. Sem. Mat. Univ. Padova, 94, 275-277 (1995) · Zbl 0852.20020
[8] Macedońska, O., On difficult problems and locally graded groups, J. Math. Sci. (NY), 142, 1949-1953 (2007) · doi:10.1007/s10958-007-0102-9
[9] Ribes, L.; Zalesskii, P., Profinite Groups (2010), Berlin: Springer, Berlin · Zbl 1197.20022
[10] Robinson, DJS, A Course in the Theory of Groups (1996), New York: Springer, New York
[11] Shumyatsky, P.: Applications of Lie ring methods to group theory. (2017). Preprint arXiv:1706.07963 [math.RA] · Zbl 0973.20027
[12] Traustason, G.: Engel groups. Groups St Andrews 2009 in Bath. Volume 2, 520-550, London Mathematical Society Lecture Note Series 388, Cambridge University Press, Cambridge (2011) · Zbl 1243.20050
[13] Wilson, JS, Two-generator conditions for residually finite groups, Bull. Lond. Math. Soc., 23, 239-248 (1991) · Zbl 0746.20018 · doi:10.1112/blms/23.3.239
[14] Wilson, JS; Zelmanov, EI, Identities for Lie algebras of pro-\(p\) groups, J. Pure Appl. Algebra, 81, 103-109 (1992) · Zbl 0851.17007 · doi:10.1016/0022-4049(92)90138-6
[15] Zelmanov, E.I.: Nil Rings and Periodic Groups. The Korean Math. Soc. Lecture Notes in Math, Seoul (1992) · Zbl 0953.16500
[16] Zelmanov, EI, On periodic compact groups, Israel J. Math., 77, 83-95 (1992) · Zbl 0786.22008 · doi:10.1007/BF02808012
[17] Zelmanov, E.I.: Lie methods in the theory of nilpotent groups. in Groups ’93 Galaway/ St Andrews, Cambridge University Press, Cambridge, pp 567-585 (1995) · Zbl 0860.20031
[18] Zelmanov, EI, Lie algebras and torsion groups with identity, J. Comb. Algebra, 1, 289-340 (2017) · Zbl 1403.17014 · doi:10.4171/JCA/1-3-2
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