$G$ and $G/H$ representations

Question

It is known that if a group $G$ has an invariant subgroup $H$ and the factor group $G/H$ has a known representation then this representation is also a representation of group $G$. But, how can we prove it? Is it enough to show that $G$ and $G/H$ are homomorphic?

Answer 1

If $\rho : G/H \to GL(V)$ is a representation of $G/H$ on the vector space $V$, and $\pi : G \to G/H$ is the natural projection ($\pi (g) = \hat g$), then $\rho \circ \pi : G \to GL(V)$ is a representation of $G$ on $V$.

Answer 2

Suppose $\rho$ is a representation of $G/H$ and you need to find a representation of $G$. Let $\rho\ :\ G/H\longrightarrow \ GL(V)$ . Then define

$\bar \rho\ : G\longrightarrow \ GL(V)$ by $\bar \rho(g)=\rho (gH)$. Check that this is a representation of $G$ and is irreducible whenever $\rho$ is irreducible.