Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 4 / e / Solution

If $N_G(D)\le H\le G$, then $N_H(D)=N_G(D)=N$ and $C_H(D)=C_G(D)=C$. Applying part (d) to $G$ and to $H$ therefore gives the same target idealCompose the two bijections. A block idempotent $e$ of $G$ with defect $D$ corresponds to the unique block idempotent $f$ of $H$ with defect $D$ satisfyingThis proves the requested form of Brauer first main theorem, including uniqueness and the equality of the Brauer images.