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有限群在特征为零的任意域上的表示的一些注记
引用本文:海进科,李正兴.有限群在特征为零的任意域上的表示的一些注记[J].数学研究与评论,2011,31(3):437-442.
作者姓名:海进科  李正兴
作者单位:青岛大学数学科学学院, 山东 青岛 266071;青岛大学数学科学学院, 山东 青岛 266071
基金项目:国家自然科学基金(Grant No.10771132),山东省自然科学基金(Grant No.Y2008A03).
摘    要: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.

关 键 词:ΓK-action  ΓK-classes  orthogonality  relations.
收稿时间:2009/4/17 0:00:00
修稿时间:2009/9/15 0:00:00

Remarks on Representations of Finite Groups over an Arbitrary Field of Characteristic Zero
Jin Ke HAI and Zheng Xing LI.Remarks on Representations of Finite Groups over an Arbitrary Field of Characteristic Zero[J].Journal of Mathematical Research and Exposition,2011,31(3):437-442.
Authors:Jin Ke HAI and Zheng Xing LI
Institution:College of Mathematics, Qingdao University, Shandong 266071, P. R. China
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.
Keywords:$\Gamma_K$-action  $\Gamma_K$-classes  orthogonality relations  
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