# Irreducible representations of groups of order $pq$: induction from normal subgroups

Consider a group $G$ of order $pq$ ($p$ and $q$ are distinct primes and also $p<q$). It is easy to show that the dimension of each irreducible representation of $G$ is $1$ or $p$. Also, it can be proved by Sylow's theorem that $G$ has the unique normal subgroup of order $q$ (moreover, if $G$ is nonabelian the subgroup of order $q$ is the unique normal subrgroup in $G$!). Besides, each irreducible representation $V$ of $G$ is a subrepresentation in $Ind_H^G(W)$ for some one-dimensional $H$-module W. However, if $\mathrm{dim}(V)>1$ then $\mathrm{dim}(V)=p$ (as it is discussed above) and $G:H=p$. It follows immediately that $\mathrm{dim}(V)=p=\mathrm{dim}(Ind_H^G(W))$
Thus we proved that each representation of $G$ is one-dimensional or induced from one-dimensional representation of the normal subgroup. The family of representations of $H$ can be discribed easily (really, $H$ is a finite cyclic group). But how to understand when $Ind_H^G(W_1)\simeq Ind_H^G(W_2)$ for $W_1,W_2\in \mathrm{Irrep}(H)$? I'll be grateful for any help!

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