Irreducible representations of group of order $pq$ There is the problem to describe dimensions of irreducible representations of a group of order $pq$, where $p$ and $q$ a distinct primes. I am doing it as follows:
Suppose $p>q$. Then by the Sylow theorem the number $N_p$ of subgroups of order $p$ divides $q$ and $N_p\equiv1 (\mathrm{mod} p)$. It means that $N_p=1$ and we get an abelian subgroup of index $q$. But the dimension of irreducible representation divides the order of a group and cannot be greater that the index of each abelian subgroup. So, the dimension $\mathrm{dim}V$ is $q$ or $1$. Is that description full? Perhaps it is possible to add something more? Perhaps we can evaluate the number of one-dimensional representations?
 A: All representations are over $\mathbb{C}$.
Theorem:  Let $G$ be a finite group, and let $Lin(G)$ denote the set of $1$-dimensional representations of $G$. Then $|Lin(G)|=[G:G']$, where $G'$ denotes the commutator subgroup of $G$.
Proof: Let $\rho$ be an irreducible representation of the abelian group $G/G'$. Then $\rho$ is $1$-dimensional. Define $\hat{\rho}:G\rightarrow \mathbb{C}^{\times}$ by $\hat{\rho}(g)=\rho(gG')$, $g\in G$. Clearly $\rho$ is a homomorphism, and hence a $1$-dimensional representation of $G$.
On the other hand, suppose that $\sigma$ is a $1$-dimensional representation of $G$. Then $G/Ker(\sigma)$ is abelian, being isomorphic to a subgroup of $\mathbb{C}^{\times}$. Thus, $G'\le Ker(\sigma)$, and hence $\sigma$ induces a group homomorphism $\overline{\sigma}:G/G'\rightarrow \mathbb{C}^{\times}$ given by $gG'\mapsto \sigma(g)$, $g\in G$. It's easily seen that the maps $Irrep(G/G')\rightarrow Lin(G), \rho \mapsto \hat{\rho}$, and $Lin(G)\rightarrow Irrep(G/G'), \sigma\mapsto \overline{\sigma}$ are mutually inverse bijections. Thus, in particular, $|Lin(G)|=|Irrep(G/G')|=|G/G'|$, as required.                                                                       QED
Now, if $G=pq$, $p>q$, then $|G'|=1$, in which case $G$ is abelian (so all irreducible representaions are $1$-dimensional), or $|G'|=p$, in which case (by the Proposition), there are precisely $q$ $1$-dimensional representations over $\mathbb{C}$.
