Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 5 / b

Regard the block algebra $RGe$ as an $R[G\times G]$-module through left and right multiplication,A defect group of a block is a p-subgroup $D\le G$ for which $\Delta D=\{(d,d):d\in D\}$ is a vertex of an indecomposable summand determining the block; equivalently, $D$ is maximal withunder the Brauer morphism. The uniqueness of vertices up to conjugacy in $G\times G$, together with the diagonal form of these vertices, shows that any two such $D$ are conjugate in $G$. Hence