Simple components and central units in group algebras

Abstract

Let G be a finite group. Berman [Dokl. Akad. Nauk 106 (1956) 767] and Witt [J. Reine Angew. Math. 190 (1952) 231] evaluate, independently, the number of simple components of the group algebra F G when F is a field of characteristic 0. In this paper we extend this result to fields of arbitrary characteristic which does not divide the order of G. We also compute the rank of the group of the central units of ZG and obtain an alternative proof of a well-known result of Ritter and Sehgal.

Central units of integral group rings

“…By construction, [U(Z |g| ) : S g ] equals the number of conjugacy classes contained in the Q-class of g. Furthermore, [S g : S g ] = 1 when g is conjugated to g −1 and [S g : S g ] = 2 when g is not conjugated to g −1 . Therefore |T g | = [U(Z |g| ) : S g ] is exactly the number of R-classes contained in the Q-class of g. Hence |B| equals the number of R-classes minus the number of Q-classes in G. By a result in [RS05,Fer04], this number coincides with the rank of Z(U(ZG)) and the proof is finished.…”

The lower central series of the unit group of an integral group ring

“…This class of groups possesses various interesting properties and is topic of active research. For instance, for a finite group G, being a cut group is equivalent to saying that the center of V(ZG) has rank zero (the rank of Z(V(ZG)) has been computed independently by Ferraz [Fer04] and Ritter-Sehgal [RS05]). For more details on finite cut groups, see ([MP18], Section 3).…”

On Idempotents and the Number of Simple Components of Semisimple Group Algebras

“…We give a formula for the rank of Z(U(RG)). Using Dirichlet's Unit Theorem one can prove that the rank of Z(U(RG)) is the difference between the number of simple components of R ⊗ Q F G and the number of simple components of F G, see [Fer04,Theorem 3.5].…”