The bounded Lie Engel property on torsion group algebras. (Q640922)

Let \(F\) be a field of characteristic \(p\neq 2\) and let \(G\) be a group with involution \(*\). Extend the involution \(F\)-linear to the group algebra \(FG\) and let \((FG)^-\) denote the set of skew elements of \(FG\), that is, \(\alpha\in(FG)^-\) if \(\alpha^*=-\alpha\). Assume \(G\) is a torsion group without elements of order \(2\). The following result is proved. The set \((FG)^-\) is bounded Lie Engel if and only if either (1) \(FG\) is bounded Lie Engel, or (2) \(p>2\), \(G\) has a \(p\)-Abelian (i.e. the commutator subgroup \(H'\) is a finite \(p\)-group) normal subgroup subgroup \(H\) of finite index and \(G\) has a normal \(*\)-invariant \(p\)-subgroup \(N\) of bounded exponent such that the induced involution on \(G/N\) is trivial. In the past two decades a lot of attention has been given on determining if Lie identities satisfied by \((FG)^-\) or by \((FG)^+\) (the symmetric elements of \(FG\)) are also satisfied by the full group algebra \(FG\). Originally the attention was focussed on the classical involution, that is \(g^*=g^{-1}\) for \(g\in G\). For a survey we refer the reader to the monograph by \textit{G. T. Lee} [Group identities on units and symmetric units of group rings. Algebra and Applications 12 (2010; Zbl 1203.16025)]. For results on arbitrary involutions one can consult work of, for example, Broche Cristo, Giambruno, Jespers, Polcino Milies, Ruiz and Sehgal.