Infinite locally soluble \(k\)-Engel groups
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 \(G \in {\mathcal E}_ 2\).
scientific article; zbMATH DE number 124184