 Green correspondence

Let $N_G(D)\le H\le G$. Set $\mathcal{X}=\{D\cap {}^{g}D:g\notin H\}$ and $\mathcal{Y}=\{H\cap {}^{g}D:g\notin H\}$. 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 $\mathcal{X}$.