= Brauer quotient of a module
{c}
{title2=$M(P)=M^P/\sum_{Q<P}\operatorname{Tr}_Q^P(M^Q)$}
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 $P$.
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