Group identities on symmetric units.
The group algebra \(FG\) of a group \(G\) over a field \(F\) is naturally endowed with an involution *, i.e., the \(F\)-linear extension of the involution on \(G\) given by \(g\mapsto g^{-1}\). The latter is called the classical involution and has received substantial interest. Of course, there are obvious more general involutions of \(FG\), namely the \(F\)-linear extensions obtained from arbitrary involutions on \(G\). Recently, there has been a surge of activity on the study of these more general involutions, see for example [\textit{O. Broche Cristo, E. Jespers, C. Polcino Milies, M. Ruiz Marín}, J. Algebra Appl. 8, No. 1, 115-127 (2009; Zbl 1192.16023)] and [\textit{E. Jespers, M. Ruiz Marín}, Commun. Algebra 34, No. 2, 727-736 (2006; Zbl 1100.16021)]. A lot of interest is given to the following topic. Let * be an involution on a ring \(R\). Let \(R^+=\{r\in R\mid r^*=r\}\), the symmetric elements of \(R\), and \(R^-=\{r\in R\mid r^*=-r\}\), the skew symmetric elements of \(R\). An important question is: what properties of \(R^+\) or \(R^-\) can be lifted to \(R\)? (see for example [\textit{I. N. Herstein}, Rings with involution. Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press (1976; Zbl 0343.16011)]). The group of units of \(R\) is denoted by \(U(R)\), the symmetric units by \(U^+(R)\). In this paper, in case \(F\) is an infinite field of \(\text{char}(F)=p\geq 0\), \(p\neq 2\) and * is an involution on a torsion group \(G\), a complete characterization is given when the *-symmetric units of \(FG\) satisfy a group identity. The main result consists of the following two parts. (1) If \(FG\) is semiprime then \(U^+(FG)\) satisfies a group identity if and only if \(G\) is Abelian or \(G\) is an SLC-group, i.e., \(G\) modulo the center is the Klein Four group. (2) If \(FG\) is not semiprime then \(U^+(FG)\) satisfies a group identity if and only if \(P\), the set of \(p\)-elements of \(G\), is a subgroup, \(FG\) satisfies a polynomial identity and one of the following holds: (i) \(G/P\) is Abelian and \(G'\) is of bounded \(p\)-power exponent; (ii) \(G/P\) is SLC and \(G\) contains a normal *-invariant \(p\)-subgroup \(B\) of bounded exponent such that \(P/B\) is central in \(G/B\) and * is trivial on \(P/B\).
scientific article; zbMATH DE number 5655722
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