number of irreps quotient by a nontrivial normal subgroup. Let $n$ be a number of irreps of the group $G$. Let $H$ be a nontrivial normal subgroup of $G$. Let $h$ be  number of irreps of the group $G/H.$ Prove that $$n>h$$
I assume that this has something to do with the fundamental theorem on homomorphisms. But I do not  manage to use it. I have no idea how to start it.
Thank you very much!
 A: It was put very well in the comment, however I will explicate further in case that was too terse as I feel that the answer to this question is very enlightening when it comes to the basics of representation theory. Given a group $G$, a normal subgroup $H \trianglelefteq G$, one has the homomorphism $\pi: G \to G/H$. Now if one has some $\rho: G/H \to GL_n(\mathbb C)$ then composing the two homomorphisms $\rho \circ \pi: G \to GL_n(\mathbb C)$ gives a representation $\hat{\rho}$ of $G$. Further, if we take $\rho$ to be irreducible then $\hat{\rho}$ is STILL irreducible because $\pi$ is an epimorphism (surjective homomorphism). 
So we have that $n \geq h$, all we need to do is show that there is some irreducible representation of $G$ that is not induced by one of $G/H$. Now there are a couple ways to do this, one is to note that there cannot be an element $g \in G$ such that $\rho(g) = I$ for all $\rho$ irreducible, and combine that with the fact that $\pi$ must have nontrivial kernel. An easier way is to note that the sum of the squares of the dimensions of the irreducible representations of a group must be the order of that group. Since all of the $\hat{\rho}$ have the same degree as $\rho$, and $G$ is bigger than $G/H$ we know that there must be more irreducible representations.
I think the first paragraph in particular provides a lot of insight into the types of proofs that show up in and intro course on the representation theory of finite groups, and moreover why this statement should be true.
