Degree 1 and 2 representations of nilpotent groups and applications to units of group rings
This is a rather technical paper concerned with the unit group \(U(ZG)\) of the integral group ring \(ZG\) of the finite group \(G\). In studying generators for these unit groups, certain components \(eQG\) of the rational group ring \(QG\) cause difficulties and are considered exceptional. These troublesome components are isomorphic to: (1) noncommutative division rings which are not totally definite quaternion algebras, (2) \(M_2(Q)\), (3) \(M_2(F)\) where \(F\) is an imaginary quadratic extension of the rationals, or (4) \(M_2(D)\) where \(D\) is a noncommutative division ring. This paper starts by classifying those finite nilpotent groups \(G\) having an exceptional component and then uses this information to describe generators for a subgroup of finite index of \(U(ZG)\).
scientific article; zbMATH DE number 784054