On the number of irreducible representations of a finite group (Q798760)

Let G be a group, K a field and r the number of irreducible representations of G over K. By a well known classical result and investigations due to \textit{R. Brauer} [Über die Darstellung von Gruppen in Galoisschen Feldern. Paris: Hermann \& Cie. (1935; Zbl 0010.34401)] r coincides with the number of classes of p-regular elements in G provided that K is a splitting field with characteristic p of G. In case that K is not a splitting field of G a general result was obtained by \textit{S. D. Berman} [Dokl. Akad. Nauk SSSR 106, 767-769 (1956; Zbl 0070.02501)], which sets the number of irreducible representations in relation to the number of those classes of p-regular elements which are defined by suitable powers of conjugates. In this note another generalization is presented following Brauer's approach while dropping the supposition of K being a splitting field for G. Call an element of G K-regular if its order is not divisible by char K. If KG denotes the group ring of G over K then \(KG/Rad KG=A_ 1\oplus...\oplus A_ r\) holds, where \(A_ i\cong Mat_{n_ i}(D_ i)\) is a matrix ring over a certain division algebra \(D_ i\) with base field K. It is proved that r and the number k of conjugacy classes of K- regular elements of G are linked by the relation \(k=\sum^{r}_{i=1}\dim_ KZ(D_ i).\) In case of a splitting field we have \(D_ i=K\) for all i and consequently \(k=r\) comes about. This contains the well known classical result in char K\(=0\) and Brauer's result in char K\(>0\).