Green correspondence with exceptional families

Under the inherited AC assumption, F2 defines $f(M)$ for every module in the $G$-domain. It has the same specified vertex $Q\in \mathcal{Z}$, so belongs to the $H$-domain. F2 also gives multiplicity one and exactly the first displayed decomposition.

F3 defines $g(U)$ for every module in the $H$-domain. It has the same vertex $Q\in \mathcal{Z}$, belongs to the $G$-domain, and satisfies multiplicity one and the second displayed decomposition.

By F4 these constructions depend only on isomorphism classes and satisfy $g(f(M))\cong M$ and $f(g(U))\cong U$. The first equality proves injectivity of $f$ and surjectivity of $g$; the second proves injectivity of $g$ and surjectivity of $f$. Thus they are inverse bijections with the decompositions and common vertex established in 1.1–1.2.

For $H=G$, F1 gives empty error families and F2–F3 give zero errors, so the assignments are identities on the stated vertex domain. If $P=1$, then $N_G(P)=G$ forces this case. Zero modules are outside both indecomposable domains; zero errors are allowed. The two families remain distinct in general, and no assertion about sources follows or is required. This completes the theorem. [F1, F2, F3, step 2.1] QED