## 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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