Descending representation on the quotient group I have this homework question in introductory representation theory:
Let $ψ : G → GL(V)$ be a representation of $G$ and let $N$ be a normal subgroup of $G$. Define $ρ : G/N → GL(V)$ by $ρ(gN) = ψ(g)$. Show that $ρ$ is a well-defined representation of $G/N$.
I think I must have $\psi$ to be trivial on $N$, for this question to make sense. Otherwise, if $gN=hN$ with $g,h$ being distinct, how can I verify $\psi(g)=\psi(h)$?

You're correct - we can't show $\rho$ is well-defined because with no hypotheses it might not be. In particular we would need $\psi(n)=1$ for all $n\in N$, which needn't be the case for all morphisms $G\to{\rm GL}(V)$ (and certainly not for all $N\trianglelefteq G$). If $\psi|_N$ does happen to be trivial we say $\psi$ descends to the representation $\rho:G/N\to{\rm GL}(V)$.
In particular, no faithful representation (i.e. injective $G\to{\rm GL}(V)$) will descend to a representation of a quotient $G/N$ when $N$ is a proper normal subgroup.
