## Finitely generated soluble groups with an Engel condition on infinite subsets.(English)Zbl 0797.20031

The authors consider the class $$E(\infty)$$ of all groups $$G$$ such that for every infinite subset $$X$$ of $$G$$ there exist distinct elements $$x,y \in X$$ such that $$[x,{_ ky}]= 1$$ for some integer $$k \geq 1$$ depending on $$X$$. If $$k$$ can be chosen to be the same for all infinite subsets of $$G$$, then $$G$$ belongs to the class $$E_ k(\infty)$$. It is proved that a finitely generated soluble group $$G$$ is an $$E(\infty)$$-group if and only if it is finite-by-nilpotent. Moreover, under the same hypotheses, $$G$$ belongs to the class $$E_ 2(\infty)$$ if and only if the subgroup $$R(G)$$ of all right 2-Engel elements of $$G$$ has finite index in $$G$$.

### MSC:

 20F45 Engel conditions 20E07 Subgroup theorems; subgroup growth 20F16 Solvable groups, supersolvable groups 20F05 Generators, relations, and presentations of groups 20F18 Nilpotent groups
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### References:

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