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Let $H \lhd G$ and let $\chi$ be a character of $H$. Let $g \in G$ and let $H^g = gHg^{-1}$.

Define $\chi^g$ to be the class function on $H^g$ given by $\chi^{g}(x) = \chi(g^{-1}xg)$.

Suppose that $\chi$ is an irreducible character of $H$ and we want to see if $Ind_{H}^{G} \chi$ is an irreducible character of $G$.

By Frobenius reciprocity, $(Ind_{H}^{G} \chi,Ind_{H}^{G} \chi)_G = (\chi, Res_{H}^{G}Ind_{H}^{G} \chi)_H$.

Let $h \in H$ and consider $(Res_{H}^{G}Ind_{H}^{G} \chi)(h) = Ind_{H}^{G}\chi(h) = \sum_{g \in G/H} \chi(g^{-1}hg) = \sum_{g \in G/H} \chi^{g}(h)$.

I have no idea how they go from $Ind_{H}^{G} \chi(h)$ to $\sum_{g \in G/H} \chi(g^{-1}hg)$? Where do cosets come in to it? Why are we summing over the cosets?

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