Modules in a block are projective relative to its defect group ($M\mid {\mathrm{Ind}}_D^G{\mathrm{Res}}_D^{G}M$)

For a block of a group algebra with identity $e$ and defect group of a block $D$, the trace criterion for defect groups writes $e={\mathrm{Tr}}_D^G(a)$ with $a\in (RG{)}^D$. On any module in the block, multiplication by $a$ is $D$-linear and its relative trace is the identity. The D. Higman criterion therefore makes the module relatively $D$-projective.