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Engel-like elements in infinite soluble groups. (English) Zbl 0684.20023
The authors consider the following sets of elements: $$A(G)=\{y|$$ $$y\in G$$; for each $$x\in G$$ there are r and d such that $$[x,_ ry]=[x,_{r+d}y]\}$$, $$A^*(G)=\{z|$$ $$z^ i\in A(G)$$ for all $$i\}$$, $$B(G)=\{x|$$ $$x\in G$$; for each $$y\in G$$ there are r and d such that $$[x,_ ry]=[x,_{r+d}y]\}$$. In analogy to Engel elements, $$x^{-1}\in A(G)$$ holds for all $$x\in B(G)$$. In a locally soluble group G the sets $$A^*(G)$$ and B(G), but not necessarily A(G), are subgroups; $$A^*(G)$$ is the unique largest locally finite-by-nilpotent normal subgroup of G and $$B(G/B(G))=1$$. If G is finitely generated and soluble and K is the subgroup generated by all finite normal subgroups of G, then B(G)/K is the hypercentre of G/K.
Reviewer: H.Heineken
##### MSC:
 20F12 Commutator calculus 20F16 Solvable groups, supersolvable groups 20E07 Subgroup theorems; subgroup growth 20F45 Engel conditions
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##### References:
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