Classification of $G$-modules Suppose that I work only on vector spaces over $\mathbb C$. If I want to classify all $n$-dimensional modules over a finite group $G$, is it enough to choose a vector space $V$ with dimension $n$ and work on $G$-modules on $V$? For example, if I want to classify all one-dimensional $G$-modules, could I just take $\mathbb C$, work on the form of my $G$-modules on $\mathbb C$, and then, since $\mathbb C$ is isomorphic to all other one-dimensional vector spaces,  just "transfer" the module structure over to any other one-dimensional vector space? Are there any $G$-module structures lost this way?
 A: Yes that's enough because, as you say, all vector spaces of a given dimension are isomorphic.  Given an arbitrary $n$-dimensional $G$-module $V$, choose a basis $\{v_1,\ldots, v_n\}$ for $V$.  Then for $g \in G$ we have $g\cdot v_i = \sum_j\rho(g)_i^j v_j$ for some $\rho(g)_i^j \in \mathbb C$.  Then the collection of $\rho(g)_i^j$ gives a map $G \to GL(n,\mathbb C)$, which is a homomorphism.  So this transfers the representation to an isomorphic representation on $\mathbb C^n$.
However, it is often useful to have a specific form of a vector space.  For example, all irreducible representations of $SU(2)$ are on homogeneous polynomials in two-variables with the action
$$
(g \cdot p)(v) = p(g^{-1} v)
$$
where $v \in \mathbb C^2$, $p$ is a homogeneous polynomial, and $g \in SU(2)$ (acting on $v$ by the defining representation).  So in this case its often more convenient to think of the $n$ dimensional irrep of $SU(2)$ as acting on homogeneous polynomials of degree $n-1$ instead of on $\mathbb C^n$.
