I'm learning about Representation Theory, and I've come across a statement I don't understand:
"If $\rho$ and \rho' are isomorphic representations, then they have the same dimension. However, the converse is not true: in $C_4$, there are four non-isomorphic 1-dimensional representations. If $\omega = e^{2 i \pi /4}$, then we have $\rho_j(\omega^i) = \omega^{ij} \ (0 \leq i \leq 3)$"
I do not understand how the $p_j$ are representations of $C_4$; as I understood it, a representation of a group was a homomorphism from the group into a group of linear transformations of vector spaces (or at least into the automorphism group of some object). But these are homomorphisms from $C_4$ into some subgroup of $C_4$. Any explanation would be appreciated.