Brauer extended first main theorem

Over a splitting field for finite group representations, blocks of a group algebra with defect group of a block $D$ correspond to $N_G(D)$-orbits of blocks $b_D$ of $k(D{C}_G(D))$ with defect $D$ whose stabilizer quotient $N_G(D,b_D)/(D{C}_G(D))$ has order coprime to $p$. Equivalently one uses the defect-zero block of $k(D{C}_G(D)/D)$. This quotient is the inertial quotient of a block.