= Nagao module theorem
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= Nagao version of Brauer second main theorem
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{synonym}
For a central idempotent $e$, a module $M=eM$, and $C_G(D)\leq K\leq N_G(D)$, put $b=\operatorname{Br}_D(e)$. Then $M\downarrow_K=bM\oplus(1-b)M$, and every indecomposable summand of the second term has vertex not containing $D$. Indeed $e-b$ is a sum of $K$-orbit sums of group elements not centralizing $D$. Each is a relative trace from a centralizer not containing $D$. Applying these traces to the module and using the local endomorphism ring of an indecomposable summand proves relative projectivity for one such subgroup.
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