Symmetric units and group identities
In the paper under review the authors discuss when the set of symmetric units of a group ring satisfies a group identity. A unit of a group algebra is called a symmetric unit if it is stable under the involution coming from the natural Hopf algebra structure of the group ring. Let \(G\) be a torsion group and let \(F\) be an infinite field of characteristic different from 2. The authors classify the groups \(G\) so that the symmetric units of the group algebra \(FG\) satisfy a group identity. The answer depends on the characteristic of the field and in case of finite, positive, characteristic, if the quaternion group \(K_8\) is a subgroup of \(G\). In more detail. The symmetric units satisfy a group identity if and only if the following holds: In the case when \(F\) is a field of characteristic 0 \(G\) is abelian or a Hamiltonian 2-group. In the case when \(F\) is of characteristic \(p\geq 3\), and \(FG\) satisfies a polynomial identity, and either if \(K_8\) is not a subgroup of \(G\) and then the derived group of \(G\) is of bounded exponent \(p^k\) for some \(k\geq 0\), or if \(K_8\leq G\) the set of \(p\)-elements of \(G\) forms a normal \(p\)-subgroup \(P\) of \(G\) so that \(G/P\) is a Hamiltonian 2-group and \(G\) is of bounded exponent \(4p^s\) for some \(s\geq 0\). The case of fields of characteristic 2 is not discussed.
scientific article; zbMATH DE number 1208703