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representation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of groups that have a finite number of elements.
==Basic definitions==
All the linear representations in this article are finite-dimensional and assumed to be complex unless otherwise stated. A representation of ''G'' is a group homomorphism ρ:''G'' → GL(''n'',C) from ''G'' to the general linear group GL(''n'',C). Thus to specify a representation, we just assign a square matrix to each element of the group, in such a way that the matrices behave in the same way as the group elements when multiplied together.
We say that ρ is a real representation of G if the matrices are real, i.e. if ρ(''G'') ⊂ GL(''n'',R).
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