Categorially equivalent and isotypic blocks (Q1330048)

Let \(p\) be a prime, and let \(D\) be a Sylow \(p\)-subgroup of a normal subgroup \(N\) of a finite group \(G\). Let \(({\mathcal O},K,k)\) be a splitting \(p\)-modular system for the subgroups of \(G\), let \(f\) be the principal block idempotent of \({\mathcal O}N\), and let \(e\) be a block idempotent of \({\mathcal O}G\) covering \(f\). Suppose that \(G = NC_ G(D)\) and that \(D\) is a defect group of \(e\). The author shows that the categories \(e{\mathcal O}G\)- mod and \(f{\mathcal O}N\)-mod are equivalent, that \(e\) is of principal type, and that there exists an isotypy between \((G,e)\) and \((N,f)\).