Spiezia, Lucia Serena A characterization of third Engel groups. (English) Zbl 0823.20038 Arch. Math. 64, No. 5, 369-373 (1995). The author gives a characterization of \(3\)-Engel groups by means of a combinatorial property. She defines the class \({{\mathcal E}_ 3}^*\) of those groups \(G\) satisfying the following condition: For every pair of infinite subsets \(X\), \(Y\) of \(G\), there exist \(x\) in \(X\) and \(y\) in \(Y\) such that \([x, y, y, y] = 1\) and proves that any infinite group in the class \({{\mathcal E}_ 3}^*\) is a third Engel group. Reviewer: L.S.Spiezia (Napoli) Cited in 1 ReviewCited in 5 Documents MSC: 20F45 Engel conditions 20E10 Quasivarieties and varieties of groups Keywords:\(3\)-Engel groups; infinite subsets; infinite groups PDFBibTeX XMLCite \textit{L. S. Spiezia}, Arch. Math. 64, No. 5, 369--373 (1995; Zbl 0823.20038) Full Text: DOI References: [1] N. D. Gupta andM. F. Newman, Third Engel groups. Bull. Austral. Math. Soc.40, 215-230 (1989). · Zbl 0675.20034 [2] H. Heineken, Engelsche Elemente der L?nge drei. Illinois J. Math.5, 681-707 (1961). · Zbl 0232.20073 [3] L.-C. Kappe andW. P. Kappe, On Three-Engel groups. Bull. Austral. Math. Soc.7, 391-405 (1972). · Zbl 0238.20044 [4] P. Longobardi, M. Maj andA. H. Rhemtulla, Infinite groups in a given variety and Ramsey’s theorem. Comm. Algebra20, 127-139 (1992). · Zbl 0751.20020 [5] B. H. Neumann, A problem of Paul Erd?s on groups. J. Austral. Math. Soc. Ser. A21, 467-472 (1976). · Zbl 0333.05110 [6] H.Neumann, Varieties of groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band37. Berlin-Heidelberg-New York 1967. · Zbl 0149.26704 [7] O. Puglisi andL. S. Spiezia, A combinatorial property of certain infinite groups. Comm. Algebra22, 1457-1465 (1994). · Zbl 0803.20024 [8] D. J. S.Robinson, Finiteness conditions and generalized soluble groups. Part I and Part II. Berlin-Heidelberg-New York 1972. · Zbl 0243.20032 [9] D. J. S.Robinson, A course in the theory of groups. Berlin-Heidelberg-New York 1982. · Zbl 0483.20001 [10] L. S. Spiezia, Infinite locally solublek-Engel groups. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9)3, 177-183 (1992). · Zbl 0791.20038 [11] L. S. Spiezia, A property of the variety of 2-Engel groups. Rend. Sem. Mat. Univ. Padova91, 225-228 (1994). · Zbl 0816.20027 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.