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A property of the variety of \(2\)-Engel groups. (English) Zbl 0816.20027

Let \(V\) be a variety of groups defined by a single law in \(n\) variables, and define \(V^*\) to be the class of groups with the property that for every set of \(n\) infinite sets \(X_ 1,\dots, X_ n\) of elements there exist elements \(x_ i \in X_ i\) that generate a subgroup in \(V\). This definition, and the question “for which varieties \(V\) is every infinite \(V^*\)-group a \(V\)-group?”, are due to P. S. Kim, A. H. Rhemtulla, and H. Smith [Houston J. Math. 17, 429-437 (1991; Zbl 0744.20033)].
The author answers the question positively for 2-Engel groups by proving the theorem: If for every pair \(X\), \(Y\) of infinite sets of elements of a group there exist elements \(x \in X\) and \(y \in Y\) such that \([x, y, y] = 1\), then the group is a 2-Engel group. The proof uses a special case of a lemma of the author’s [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 3, No. 3, 177-183 (1992; Zbl 0791.20038)], namely that in a group of the theorem the centraliser of every element is infinite.

MSC:

20E10 Quasivarieties and varieties of groups
20F45 Engel conditions
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References:

[1] P.S. Kim - A.H. Rhemtulla - H. Smith , A characterization of infinite metabelian groups , Houston J. Math. , 17 ( 1991 ), pp. 429 - 437 . MR 1126607 | Zbl 0744.20033 · Zbl 0744.20033
[2] P. Longobardi - M. Maj - A.H. Rhemtulla , Infinite groups in a given variety and Ramsey’s theorem , Commun. Algebra , 20 ( 1982 ), pp. 127 - 139 . MR 1145329 | Zbl 0751.20020 · Zbl 0751.20020
[3] B.H. Neumann , A problem of Paul Erdös on groups , J. Austral. Math. Soc. , 21 ( 1976 ), pp. 467 - 472 . MR 419283 | Zbl 0333.05110 · Zbl 0333.05110
[4] A.H. Rhemtulla - H. Smith , On infinite locally finite groups and Ramsey’s theorem , Atti Accad. Naz. Lincei Rend. Cl. Fis. Mat. Natur. , ( 9 ) 3 ( 1992 ), pp. 177 - 183 . MR 1186913
[5] D.J.S. Robinson , Finiteness Conditions and Generalized Soluble Groups, Part I and Part II , Springer-Verlag , Berlin , Heidelberg , New York ( 1972 ). Zbl 0243.20033 · Zbl 0243.20033
[6] L.S. Spiezia , Infinite locally soluble k-Engel groups , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. , to appear. MR 1186913 | Zbl 0791.20038 · Zbl 0791.20038
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