A theorem on units of integral group rings
Let G be a finite group and \({\mathcal C}\) the family of cyclic subgroups of G. Let \(U=U({\mathbb{Z}}G)\) be the unit group of the integral group ring \({\mathbb{Z}}G\). Let H be the normal subgroup of U generated by the unit groups \({\mathbb{Z}}C^ x, C\in {\mathcal C}\). The author proves that, under certain restrictions on G, H has finite index in U. A theorem of Bass concerning \(K_ 1({\mathbb{Z}}G)\) and a theorem of Margulis on discrete subgroups of Lie groups have a major role in the paper. The conditions imposed on G are shown to guarantee that all the simple components of \({\mathbb{Q}}G\) coming from a 2-dimensional representation of G are of the following form: (1) a totally definite quaternion algebra; (2) \(M_ 2(K)\), where the algebraic number field K has infinitely many units; (3) central over a totally complex field with infinitely many units. Corollary 1.3. If G has odd order, then H has finite index in U.
scientific article; zbMATH DE number 4031889
scientific article