= Block idempotents centralize a normal p-subgroup
For a normal <p-subgroup> $P\triangleleft G$, every <central idempotent> $e$ of $kG$ satisfies $e=\operatorname{Br}_P(e)\in kC_G(P)$. Indeed a nonfixed conjugation orbit sum maps to zero in $k(G/P)$, so $e-\operatorname{Br}_P(e)$ 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 $kC_G(P)$.
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