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.
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 $V$.