Spiezia, Lucia Serena Infinite locally soluble \(k\)-Engel groups. (English) Zbl 0791.20038 Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 3, No. 3, 177-183 (1992). Summary: We deal with the class \({\mathcal E}^*_ k\) of groups \(G\) for which whenever we choose two infinite subsets \(X\), \(Y\) there exist two elements \(x \in X\), \(y \in Y\) such that \([x,\underbrace{y,\dots,y}_ k]= 1\). We prove that an infinite finitely generated soluble group in the class \({\mathcal E}^*_ k\) is in the class \({\mathcal E}_ k\) of \(k\)-Engel groups. Furthermore, with \(k = 2\), we show that if \(G \in {\mathcal E}^*_ 2\) is an infinite locally soluble or hyperabelian group then \(G \in {\mathcal E}_ 2\). Cited in 1 ReviewCited in 3 Documents MSC: 20F45 Engel conditions 20F19 Generalizations of solvable and nilpotent groups 20E25 Local properties of groups 20E10 Quasivarieties and varieties of groups 20E34 General structure theorems for groups Keywords:infinite subsets; finitely generated soluble group; \(k\)-Engel groups; infinite locally soluble; hyperabelian group PDFBibTeX XMLCite \textit{L. S. Spiezia}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 3, No. 3, 177--183 (1992; Zbl 0791.20038) Full Text: EuDML