Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 5 / a / Solution

Let $D$ be a $p$-subgroup and let $N_G(D)\le H\le G$. PutA 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 thatThe 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.