Lie properties of symmetric elements in group rings. II.
Let \(F\) be a field of characteristic different from \(2\), and \(G\) a group with involution *. Let \((FG)^+\) denote the set of symmetric elements of \(FG\), i.e. the elements \(\alpha\in FG\) so that \(\alpha=\alpha^*\). In part I [J. Algebra 321, No. 3, 890-902 (2009; Zbl 1169.16014)] \textit{A. Giambruno, C. Polcino Milies}, and \textit{S. K. Sehgal} showed that if \(G\) has no \(2\)-elements, and \((FG)^+\) is Lie nilpotent (resp. Lie \(n\)-Engel), then \(FG\) is Lie nilpotent (resp. Lie \(m\)-Engel, for some \(m\)). In this paper the authors complete this result by classifying the groups containing \(2\)-elements such that \((FG)^+\) is Lie nilpotent or Lie \(n\)-Engel. The main results are as follows. Let \(F\) be a field of characteristic \(p>2\), and let \(G\) be a group with involution *. Theorem 1. Suppose that \(FG\) is not Lie nilpotent. Then \((FG)^+\) is Lie nilpotent if and only if \(G\) is nilpotent, and \(G\) has a finite normal *-invariant \(p\)-subgroup \(N\) such that \(G/N\) is an SLC-group. By definition, a group \(H\) endowed with an involution * is an SLC-group if \(H/Z(H)\) is the Klein four group (and thus \(G\) has a unique nonidentity commutator, say \(z\)) and \(g^*=g\) if \(g\in Z(G)\), and otherwise \(g^*=zg\). Theorem 2. Suppose that \(FG\) is not Lie \(m\)-Engel, for any \(m\). Then \((FG)^+\) is Lie \(n\)-Engel if and only if \(G\) is nilpotent, \(G\) has a \(p\)-Abelian *-invariant normal subgroup \(A\) of finite index, and \(G\) has a normal *-invariant \(p\)-subgroup \(N\) of bounded exponent, such that \(G/N\) is an SLC-group.
scientific article; zbMATH DE number 5534986