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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 \(S\).

MSC:

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