Right Engel elements of stability groups of general series in vector spaces. (English) Zbl 1428.20036

Summary: Let \(V\) be an arbitrary vector space over some division ring \(D\), \(\mathbf{L}\) a general series of subspaces of \(V\) covering all of \(V\;\{0\}\) and \(S\) the full stability subgroup of \(\mathbf{L}\) in \(\operatorname{GL}(V)\). We prove that always the set of bounded right Engel elements of \(S\) is equal to the \(\omega\)-th term of the upper central series of \(S\) and that the set of right Engel elements of \(S\) is frequently equal to the hypercentre of \(S\).


20F45 Engel conditions
20F19 Generalizations of solvable and nilpotent groups
20H25 Other matrix groups over rings
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