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What is meant by dimension of a representation in the following excersize: "Prove that any irreducible representation of an abelian group has the dimension of 1"? I looked at the solution, and it proves that any irreducible representation of an abelian group is scalar. I understand the proof, but I still can't figure out what is meant by dimension.

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A representation of a group $G$ is a vector space $V$ together with an action of $G$ on $V$ by linear transformations. The dimension of the representation is just the dimension of $V$.

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    I mean, given that Chris has just defined the dimension of a representation to be the dimension of the corresponding vector space, the exercise and your interpretation of it are tautologically the same. Anyway, the answer to your question is 'yes' :)2011-04-10