Green correspondence for modules of vertex exactly p

By F2, every $P\cap {}^{s}P$ with $s\mathrm{\notin }H$ has order smaller than $\mathrm{\mid }P\mathrm{\mid }$. No conjugate of $P$ can be contained in it. Thus the fixed subgroup $P$ belongs to the admissible class $\mathcal{Z}$.

Apply F1 under A1 to modules with this vertex. In each direction the distinguished module has the same vertex $P$, so both maps preserve the fixed-vertex subclasses. Their two inverse identities remain valid on these subclasses, proving the required bijection. The error decompositions remain respectively $\mathcal{Y}$- and $\mathcal{X}$-projective as in F1, with no equality between the families asserted. If $P=1$, then $H=G$; if $H=G$ directly, the correspondence is the identity and the errors are zero. Nonzero indecomposable inputs and the inherited AC assumption are unchanged.