Fong–Reynolds matrix decomposition

Let $N◃G$, let $b$ be a block of a group algebra of $kN$, and let $I$ be its inertia subgroup. A block of $kG$ covering $b$ is a full matrix algebra of degree $[G:I]$ over its corresponding block of $kI$. When $I=N$, the corner is just $kNb$. Let $e_B$ be the global block idempotent and put $c=e_{B}b$. With coset representatives $g_i$, the elements $g_{i}c{g}_j^{-1}$ are matrix units in that global block because different conjugates of $b$ are orthogonal. When $I=N$, one has $c=b$.