On a Theorem of Barker–Puig that Refines Alperin’s Weight Conjecture for Blocks in p-Solvable Finite Groups via Clifford Theory

We present a much shorter Clifford Theoretic Proof of an important Theorem of Barker and Puig on a refinement of Alperin’s Weight Conjecture for Blocks in p-solvable finite groups. Our proof employs a standard Clifford Theoretic approach. We also demonstrate a relationship between the Green correspondence and Sibley’s concept of a vertex pair. Consequently the main theorem can be stated in terms of Sibley’s vertex pairs.