# Induced characters of $G$ from a normal subgroup $H$

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?