| Remarks on Representations of Finite Groups over an Arbitrary Field of Characteristic Zero |
| Received:April 17, 2009 Revised:September 15, 2009 |
| Key Words:
$\Gamma_K$-action $\Gamma_K$-classes orthogonality relations.
|
| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10771132) and the Natural Science Foundation of Shandong Province (Grant No.Y2008A03). |
|
| Hits: 2562 |
| Download times: 1947 |
| Abstract: |
| Let $G$ be a finite group and $K$ a field of characteristic zero. It is well-known that if $K$ is a splitting field for $G$, then $G$ is abelian if and only if any irreducible representation of $G$ has degree 1. In this paper, we generalize this result to the case that $K$ is an arbitrary field of characteristic zero (that is, $K$ need not be a splitting field for $G$), and we also obtain the orthogonality relations of irreducible $K$-characters of $G$ in this case. Our results generalize some well-known theorems. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2011.03.007 |
| View Full Text View/Add Comment |
