Grothendieck groups of dihedral and quaternion group rings
In this interesting paper the author gives a description of the Grothendieck group \(G_ 0({\mathbb{Z}}G)\), where G is the semidirect product of a cyclic group C with a finite group H, in terms of certain twisted group rings of H, by extending the results of \textit{H. W. Lenstra} [ibid. 20, 173-193 (1981; Zbl 0467.16016)] from G abelian to the present situation. This is then applied to explicitly describe \(G_ 0({\mathbb{Z}}G)\) for G dihedral groups and generalized quaternion groups. For example for \(G=D_{2n}\) the dihedral group of order 2n let for \(d| n\), \(\zeta_ d\) be a primitive dth root of unity. \({\mathbb{Q}}[\zeta_ d]_+\) the maximal real subfield of \({\mathbb{Q}}[\zeta_ d]\) with ring of integers \({\mathbb{Z}}[\zeta_ d]_+\) and \({\mathbb{Z}}[\zeta_ d,1/d]_+=:{\mathbb{Z}}[\zeta_ d]_+[1/d]\). Then \[ G_ 0({\mathbb{Z}}D_{2n})={\mathbb{Z}}^{(\epsilon)}\oplus \oplus_{d| n}({\mathbb{Z}}\oplus Cl({\mathbb{Z}}[\zeta_ d,1/d]_+)) \] where \(\epsilon =2\) for n odd and \(\epsilon =4\) for n even, and Cl denotes the class-group. Finally, it is shown that for a finite group G with no commuting elements of relatively prime order, the ''restriction of scalars'' map \(G_ 0(\Gamma_ G)\to G_ 0({\mathbb{Z}}G)\), where \(\Gamma_ G\) is a maximal order containing \({\mathbb{Z}}G\), is an isomorphism, a result suggested by S. Chase.
scientific article; zbMATH DE number 3889776
scientific article