I have a (probably very straightforward) question about blocks and simple modules. The problem I'm having is on p103 of Local representation theory by JL Alperin.
Let $G$ be a finite group. Let $B$ be a block of $G$ with defect group $D$. Let $b$ be the block of $N_G(D)$ which is the Brauer correspondent of $B$. Let $S$ be a simple $kN_G(D)$-module lying in $b$.
Apparently, $S$ must be a $k[N_G(D)/D]$-module, but I don't see this. Why is this true?