What is the dimension of a representation

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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1 Answer

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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So should I interpret the exceresize as "Prove that an abelian group can only have irreducible representation on a vector space with dimension of 1"? – Alexei Averchenko Apr 10 '11 at 13:08
Well, yes, that's a tautology now. – wildildildlife Apr 10 '11 at 14:38

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@wild: why is that a tautology now? – Rasmus Apr 10 '11 at 14:47
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' :) – wildildildlife Apr 10 '11 at 18:43
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