= Green correspondence
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Let $N_G(D)\leq H\leq G$. Set $\mathcal X=\{D\cap{}^gD:g\notin H\}$ and $\mathcal Y=\{H\cap{}^gD:g\notin H\}$. <Green correspondence> pairs indecomposable modules with vertex $D$ for $G$ and $H$: restriction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal Y$; induction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal X$. The two assignments are inverse and preserve sources. Its more general form applies to vertices contained in $D$ but not conjugate into any member of $\mathcal X$.
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