 Brauer third main theorem

If a block of a group algebra $b$ of $kH$ has defect group of a block $D$ and $D{C}_G(D)\le H\le G$, then its block induction $b^G$ is the principal block if and only if $b$ is the principal block. Equivalently, the Brauer morphism of the principal block idempotent at any p-subgroup $Q$ is the principal block idempotent of $k{C}_G(Q)$.