= Green correspondence from Mackey multiplicity
{c}
Let $N_G(D)\leq H\leq G$ and let $U$ be indecomposable with vertex $D$ over $H$. The <Mackey restriction formula> writes $\operatorname{Res}_H^G\operatorname{Ind}_H^GU=U\oplus T$, where the vertices in $T$ lie in $H\cap{}^gD$ for $g\notin H$. None of these subgroups contains an $H$-conjugate of $D$: equality would put $g$ in $N_G(D)H=H$. Every summand of the induction with a $G$-conjugate of vertex $D$ contributes a vertex-$D$ summand on restriction. <Krull-Schmidt decomposition> therefore forces exactly one such summand, with multiplicity one. Choosing sources and using transitivity of induction proves the inverse correspondence and preservation of sources.
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