= Solution
Let $D$ be a $p$-subgroup and let $N_G(D)\le H\le G$. Put
$$
\mathcal X=\{D\cap{}^gD:g\in G\setminus H\},\qquad
\mathcal Y=\{H\cap{}^gD:g\in G\setminus H\}.
$$
A <module> relatively projective for a family means a summand of a sum of <modules> induced from its members, allowing <subgroups> and conjugates. The <Green correspondence> gives inverse bijections $f,g$ between the indecomposable $RG$-modules and $RH$-modules with <module vertex> $D$, up to isomorphism, such that
$$
\operatorname{Res}_H^G M\cong f(M)\oplus M_{\mathcal Y},\qquad
\operatorname{Ind}_H^G U\cong g(U)\oplus U_{\mathcal X}.
$$
The error terms are respectively relatively $\mathcal Y$-projective and $\mathcal X$-projective. In particular the displayed correspondents are the unique summands of <module vertex> $D$, occur with multiplicity one, and have the same <module source> as the original <module>, up to <normalizer> conjugacy.
We will also use the general form of <Green correspondence>: replace <module vertex> $D$ by any <module vertex> $Q\le D$ not conjugate into a member of $\mathcal X$. Restriction and induction again have unique summands with that <module vertex>, and the same error-family description. This is important in the final induction argument: merely knowing that an error term has a proper <module vertex> would not give the required intersection bound.
Back to article page
