Projective modules over a finite group algebra are injective
= Projective modules over a finite group algebra are injective
For finite-dimensional modules over $kG$, projectivity and injectivity are equivalent. Indeed, $(kG)^*$ is injective because $\operatorname{Hom}_{kG}(-,(kG)^*)\cong\operatorname{Hom}_k(-,k)$ is exact, while symmetry gives $(kG)^*\cong kG$.