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Bounded Engel elements in residually finite groups satisfying an identity.
(2021) Bastos Júnior, Raimundo de Araújo; Silveira, Danilo Sanção da
Let G be a m-generator residually finite group. We show that: 1) if every ck-value in G is right n-Engel, then ckðGÞ is s-Engel for some fk, m, ng-bounded number s; 2) if G satisfies an identity w 1 and G can be generated by a commutator-closed set of left n-Engel elements, then G has {m, n, w}-bounded class. 3) if G has a specific generating set X in which some power of each element in X is a bounded left Engel, then G is virtu- ally nilpotent.
On residually finite groups satisfying an Engel type identity.
(2020) Silveira, Danilo Sanção da
Let n, q be positive integers. We show that if G is a finitely generated residually finite group satisfying the identity [x,n yq ] ≡ 1, then there exists a function f (n) such that G has a nilpotent subgroup of finite index of class at most f (n). We also extend this result to locally graded groups.