Engel-like elements in infinite soluble groups. (English) Zbl 0684.20023

Brandl, Rolf R. A.; Brookes, Christopher J. B.

Proc. Edinb. Math. Soc., II. Ser. 32, No. 3, 337-343 (1989).

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

Keywords:

commutator cycles; Engel elements; locally soluble group; locally finite- by-nilpotent normal subgroup; finite normal subgroups; hypercentre

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