Let $G$ is an infinite group with two representations $\rho: G \rightarrow GL(V)$ and $\tau: G \rightarrow GL(W)$ for finite-dimensional vector spaces $V, W$ over $\mathbb{C}$. Let $\chi_{\rho}$ and $\chi_{\tau}$ be the corresponding characters for the rep's $\rho, \tau$.
Q1: If $G$ is finite, it is known that if $\chi_{\rho} = \chi_{\tau}$ then $V \cong W$. Can we say the same for infinite $G$?
Q2: What if the characters are equal up to a constant multiple, say $\chi_{\rho} = 2\chi_{\tau}$? Can we still relate the spaces $V$ and $W$ somehow? Or at least can we "extend" the pair $(G, \tau, W)$ (whatever that means) to get equivalent characters?
I'm new to representation theory and am reading some notes currently, but I could not find any information on this particular problem.