Grothendieck groups of dihedral and quaternion group rings

Abstract

For a finite group G, let G 0( Z G) denote the Grothendieck group of finitely-generated Z G-modules with relations arising from short-exact sequences. It is shown that the methods of Lenstra for G Abelian can be adapted to semidirect products G = π ⋊ Γ with π cyclic, resulting in a description of G 0( Z G) as a direct sum of Grothendieck groups of certain twisted group rings. In the case of the dihedral groups D 2 n and the generalized quaternion groups Q 4 m , this leads to explicit formulas. Finally, it is shown that for a group G with no commuting elements of relatively-prime orders, the restriction-of-scalars map G 0( M G) → G 0( Z G), where M G is a maximal order containing Z G, is an isomorphism, a result suggested by S. Chase.