 Brauer quotient of a module

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 $P$.