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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.
Generalized torsion elements in groups.
(2023) Bastos Júnior, Raimundo de Araújo; Schneider, Csaba; Silveira, Danilo Sanção da
A group element is called a generalized torsion element if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the generalized order of an element in a finite group G using its character table.