Clifford correspondence (${\mathrm{Ind}}_T^G:\mathrm{Irr}(T\mid \theta )\stackrel{\sim }{\to }\mathrm{Irr}(G\mid \theta )$)

For $N◃G$, $\theta \in \mathrm{Irr}(N)$ and $T=I_G(\theta )$, induction gives the displayed bijection. The inverse takes the $\theta$-isotypic component of an irreducible $G$-module, viewed as a $T$-module. Induction produces distinct $N$-isotypic components indexed by $G/T$. Any nonzero $G$-submodule meets one of them and hence, by translation and irreducibility over $T$, contains them all. The same argument shows that the inverse component is irreducible and determines the original inducing module uniquely.