= Clifford correspondence
{c}
{title2=$\operatorname{Ind}_T^G:\operatorname{Irr}(T\mid\theta)\xrightarrow{\sim}\operatorname{Irr}(G\mid\theta)$}
For $N\triangleleft G$, $\theta\in\operatorname{Irr}(N)$ 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.
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