Star-group identities and groups of units.
This paper is an extension of the authors' [J. Algebra 322, No. 8, 2801-2815 (2009; Zbl 1193.16027)]. Analogously to a star identity in rings with involution a star identity in groups is defined as an element of the free group of countable rank with involution evaluating identically to 1 for elements of a group \(G\) with involution. Suppose that \(G\) is a torsion group with involution * and that \(F\) is an infinite field with \(\text{char\,}F\neq 2\). Extend * linearly to \(FG\). The main result is the following. (1) If \(FG\) is semiprime then \(U\) satisfies a star 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 \(U\) satisfies a star 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 5833787