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Brauer's first main theorem gives a bijection between the blocks of $kG$ with defect group $D$ and the blocks of $k{N}_G(D)$ with defect group $D$. Corresponding blocks are related by their images under the Brauer morphism.

Let $D$ be a p-subgroup of $G$, put $N=N_G(D)$, and let $k$ have characteristic $p$. The Brauer morphism intertwines the two relative traces:Indeed, $D$ acts on $G/D$ by left multiplication. A coset $gD$ is fixed exactly when $g\in N$, and every other orbit has size divisible by $p$. After applying ${\mathrm{Br}}_D$, the summands belonging to one such orbit are equal, so every nonfixed orbit contributes zero in characteristic $p$; the fixed cosets give the trace from $D$ to $N$.

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

Let $D$ act on the group algebra $kG$ by conjugation. Its fixed-point algebra isand the centralizer $C_G(D)$ consists of the elements of $G$ commuting with every element of $D$. The Brauer morphism isThus $\beta$ deletes the coefficients of basis elements outside $C_G(D)$. The nonfixed $D$-orbits have cardinality divisible by $p$, so the usual orbit argument shows that this projection is a unital ring homomorphism. Hence