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This is true for finite groups, and it is a consequence of the non-singularity of the Cartan matrix (whose determinant is a power of $p$) in the algebraically closed case. The Cartan invariant $c_{ij}$ gives the multiplicity of the $j$-th simple module as a composition factor of the $i$-th projective indecomposable. If there were two non-isomorphic projective indecomposables with the same composition factors, the Cartan matrix would certainly be non-singular. I believe the result may have been stated by R. Swan.