= Brauer morphism and relative trace
{c}
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>:
$$
\operatorname{Br}_D\!\left(\operatorname{Tr}_D^G(a)\right)
=\operatorname{Tr}_D^N\!\left(\operatorname{Br}_D(a)\right)
\qquad(a\in(kG)^D).
$$
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 $\operatorname{Br}_D$, 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 $N$.
Back to article page
