\(*\)-group identities on units of group rings. (Q2856432)

Analogous to \(*\)-polynomial identities in rings, the concept of \(*\)-group identities in groups is introduced. The author gives the history and motivation for investigating \(*\)-identities in unit groups of group algebras. Some recent results are surveyed.NEWLINENEWLINE The main result in the paper is due to \textit{A. Giambruno} et al. [Arch. Math. 95, No. 6, 501-508 (2010; Zbl 1216.16024)] and is the following. Let \(F\) be an infinite field of characteristic not \(2\), \(G\) a torsion group and \(*\) an involution on \(G\) extended linearly to the group algebra \(FG\). The following conditions are equivalent: (1) the symmetric units \(u\) (i.e. \(u=u^*\)) of \(FG\) satisfy a group identity; (2) the units of \(FG\) satisfy a \(*\)-group identity; (3) one of the conditions holds; (a) \(FG\) is semiprime and \(G\) is Abelian or SLC (i.e. \(G\) has a unique non-identity commutator \(z\) and, for all \(g\in G\), one has \(g^*=g\) if \(g\) is central and \(g^*=gz\) otherwise), (b) \(FG\) is not semiprime, the \(p\)-elements of \(G\) form a normal subgroup \(P\), \(G\) has a \(p\)-Abelian normal subgroup of finite index, and either \(G'\) is a \(p\)-group of bounded exponent, or \(G/P\) is an SLC-group and \(G\) contains a normal \(*\)-invariant \(p\)-subgroup \(B\) of bounded exponent such that \(P/B\) is central in \(G/B\) and the induced involution acts as the identity on \(P/B\).NEWLINENEWLINE For a longer survey on the topic, the reader is referred to \textit{G. T. Lee} [Commun. Algebra 40, No. 12, 4540-4567 (2012; Zbl 1266.16042)].