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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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