Brauer morphism (${\mathrm{Br}}_D$)

For a p-subgroup $D\le G$, let $D$ act on $kG$ by conjugation. The Brauer morphism is the algebra homomorphism

For a normal p-subgroup $P◃G$, every central idempotent $e$ of $kG$ satisfies $e={\mathrm{Br}}_P(e)\in k{C}_G(P)$. Indeed a nonfixed conjugation orbit sum maps to zero in $k(G/P)$, so $e-{\mathrm{Br}}_P(e)$ belongs to the nilpotent ideal $J(kP)kG$. Normality makes the Brauer image a central idempotent of $kG$, and two central idempotents congruent modulo a nilpotent ideal coincide. Consequently the blocks of a group algebra of $kG$ correspond to $G$-orbits of blocks of $k{C}_G(P)$.

For a $p$-subgroup $P$, the Brauer quotient removes all fixed vectors produced by proper-subgroup relative traces. On a permutation module it has a basis of the permutation-basis elements fixed by $P$: nonsingleton orbit sums are traces and disappear. The construction commutes with direct sums and idempotent projections. For an indecomposable trivial source module, it is nonzero exactly when a vertex contains a conjugate of $P$.

For $N=N_G(D)$ and $C=C_G(D)$, the Brauer morphism restricts to a surjectionThis follows from Brauer morphism and relative trace and the fact that $kC$ is contained in $(kG{)}^D$. The target is an ideal in the commutative algebra $(kC{)}^N$. Applying primitive idempotents under a surjection of commutative Artinian ideals and the trace criterion for defect groups identifies block idempotents with defect group $D$ with the primitive idempotents of $J_D$. Applying the same argument to $N$ gives Brauer first main theorem. The whole kernel need not be nilpotent, since blocks with smaller defect can be killed.

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 $N$.