ABSTRACT
For a ring R with unity 1, we denote by U(R) the unit group of R, i.e., the group of invertible elements in R. For decades one has investigated the unit group of a group ring R[G] of a finite group G over a (commutative) ring R. This is a very important ring, mainly because the representations of G on R-modules can be considered as R[G]-modules. The unit group U(R[G]) plays an important role in the study of the algebraic structure of G versus the ring structure of its group ring R[G]. This for example is shown via the study of the isomorphism problem for integral group rings. Hence integral group rings are very interesting algebraic structures because of their obvious relationship with group theory and ring theory and because the investigations in the structure also involve, for example, the theory of fields, linear algebra, algebraic topology, algebraic number theory, and algebraicKtheory. Thus the research in group rings is a subject where many branches of algebra meet and the knowledge of the unit group is crucial herein. A reasonable approach to study this problem is considering Z[G] as an
order (we give a formal definition later in the text) in the rational group algebra Q[G]. This idea comes from the important commutative case where the ring of algebraic integers Ok in an algebraic number field k is an order for which the unit group is described in the following theorem (see for example [31, Theorem IV.3.13]).
