Irreducible representation over splitting field surjects to matrix ring

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Statement

Suppose G is a finite group and k is a splitting field for G. Suppose \rho:G \to GL(n,k) is an irreducible representation for G over k. Then, the map extends uniquely by k-linearity to a k-linear map from the group ring to the matrix ring:

\tilde{\rho}: kG \to M(n,k)

The claim is that \tilde{\rho} is surjective.

Instead of requiring k to be a splitting field, we can require only that k have characteristic not dividing the order of G and the representation \rho be absolutely irreducible.

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