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That is -- is it true that if projective $k[G]$-modules have same composition factors then they are isomorphic?

This is easy to see for $\text{char}(k)=0$, or if $G$ is a composition of a $p$-group and a $p'$-group. Serre in "Linear Representations of Finite Groups" (a remark in 16.2 after Corr.2) states this as a well-known fact: "Indeed we know that the equality $[P] = [ P']$ (...) is equivalent to $P = P'$)". But unfortunately no references.

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