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Infinite locally soluble \(k\)-Engel groups. (English) Zbl 0791.20038

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 \(G \in {\mathcal E}_ 2\).

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
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