On the principal blocks of finite groups with Abelian Sylow \(p\)-subgroups (Q5936196)

Let \(K\) be an algebraically closed field of prime characteristic \(p\), and let \(B\) be a \(p\)-block of a finite group \(G\). Denote by \(\ell(B)\) the number of isomorphism classes of irreducible \(KG\)-modules in \(B\). Pursuing the well-known weight conjectures of J.~Alperin, the authors show that if \(G\) has an Abelian Sylow \(p\)-subgroup \(P\) such that \(|N_G(P)/C_G(P)|\) is a prime \(t\), then \(\ell(B(G))=t\), where \(B(G)\) is the principal block.