Projective modules over a finite group algebra are injective

For finite-dimensional modules over $kG$, projectivity and injectivity are equivalent. Indeed, $(kG{)}^{\ast }$ is injective because ${\mathrm{Hom}}_{kG}(-,(kG{)}^{\ast })\cong {\mathrm{Hom}}_k(-,k)$ is exact, while symmetry gives $(kG{)}^{\ast }\cong kG$.

If $P$ is an indecomposable projective $kG$-module, then its head and socle are simple and naturally isomorphic:This is the identity Nakayama permutation of the symmetric algebra $kG$.

For every simple finite-dimensional $kG$-module $S$,

Let $M$ be an indecomposable finite-dimensional $kG$-module and let $P_k$ be the projective cover of the trivial module. Then