Exercise 1, § 21

Let A be a commutative ring, G a group, and H a subgroup of G of finite index. Suppose that the element $(G : H)1_A$ of A is invertible. Let M be an A[G]-module and N an A[G]-submodule of M. Prove that if N admits a supplementary A[H]submodule in M, then it admits a supplementary A[G]-submodule in M (adapt the proof of Theorem 1 of VIII, p. 401). Deduce that an A[G]-module that is projective as an A[H]-module is projective.