Just non hypercentral elements and the Bell-centre (Q915860)

For every group G and every word w, a w-centre of G can be defined that coincides with the centre of G if \(w=[X,Y]\) is the commutator word. For the n-Bell word \(w=[X^ n,Y]^{-1}\cdot [X,Y^ n]\) [see \textit{H. E. Bell}, Arch. Math. 24, 34-38 (1973; Zbl 0251.16021) and \textit{R. Brandl} and \textit{L.-C. Kappe}, Commun. Algebra 17, No.4, 787-807 (1989; Zbl 0672.20019)], the elements of the corresponding n-Bell centre of a soluble group are left Engel elements if both n and n-1 are prime powers. Examples show that the hypothesis on n is necessary. In the finite case, analogous results on the embedding of the n-Bell centres into the hypercentre of G are deduced.