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Two conditions for infinite groups to satisfy certain laws. (English) Zbl 1031.20018

P. Longobardi, M. Maj and A. H. Rhemtulla posed the following question: Let \(w=w(x_1,\dots,x_n)\) be a group word in \(n\) variables \(x_1,\dots,x_n\), and \(V(w)\) the variety of groups defined by the law \(w=1\). Let \(V(w^\#)\) be the class of all groups \(G\) in which for every \(n\) infinite subsets \(S_1,\dots,S_n\) there exist \(s_i\in S_i\) such that \(w(s_1,\dots,s_n)=1\). Is there some word \(w\) and an infinite group \(G\) such that \(G\in V(w^\#)\) but \(G\not\in V(w)\)? (see [Commun. Algebra 20, No. 1, 127-139 (1992; Zbl 0751.20020)] and also question 15.1 of The Kourovka notebook. Unsolved problems in group theory. Transl. from the Russian. Including archive of solved problems. [15th augm. ed. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2002; Zbl 0999.20002)]).
The answer to this question is likely to be “yes”. It is known that \(w\) cannot be any of the words \([x_1,\dots,x_n]\) [P. Longobardi, M. Maj and A. H. Rhemtulla, loc. cit.]; \([x_1,x_2]^2\) [P. Longobardi and M. Maj, Houston J. Math. 19, No. 4, 505-512 (1993; Zbl 0813.20026)]; \((x_1x_2)^3x_2^{-3}x_1^{-3}\) [A. Abdollahi, Arch. Math. 73, No. 2, 104-108 (1999; Zbl 0936.20038)]; \(x_1^{a_1}\cdots x_n^{a_n}\) where \(a_1,\dots,a_n\) are arbitrary non-zero integers [A. Abdollahi and B. Taeri, Rend. Semin. Mat. Univ. Padova 104, 129-134 (2000; Zbl 1013.20021)]; \((x_1x_2)^2(x_2x_1)^{-2}\) and \([x_1^n,x_2]\) where \(n\in\{3,6\}\cup\{2^k\mid k\in\mathbb{N}\}\) [A. Abdollahi and B. Taeri, Bull. Malays. Math. Soc., II. Ser. 22, No. 1, 87-93 (1999; Zbl 1006.20034)].
Also the above question has been studied on various classes of groups (see [G. Endimioni, Commun. Algebra 23, No. 14, 5297-5307 (1995; Zbl 0859.20021)], [A. Abdollahi, Bull. Aust. Math. Soc. 62, No. 1, 141-148 (2000; Zbl 0964.20019) and A combinatorial problem in infinite groups. Bull. Malays. Math. Sci. Soc. (2) 25, No. 2, 101-114 (2002)]).
In the paper under review the author proves that in the above question \(w\) cannot be the words \([x_1^m,x_2^m]\) and \((x_1^mx_2^m\cdots x_n^m)^2\) where \(m\in\{2^k\mid k\geq 0\}\).

MSC:

20E10 Quasivarieties and varieties of groups
20F12 Commutator calculus
20F05 Generators, relations, and presentations of groups
20F45 Engel conditions
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