I'm learning serre's famous galois cohomology. and on page 12 he gives 2 exercises. it seems that I need the following fact in finite group cohomology:
$f: G_1 \rightarrow G_2 $ is surjective if and only if for any $G_2$ -module $A$, the induced map $f*: H^1(G_2, A) \rightarrow H^1(G_1, A)$ is injective on the cohomology level.
suppose $f$ surjective, and from the definition of the induced $G_1$ module strucutre on $A$, I obtain the injectivity of the cohomology. I try to prove the converse, but met a problem. I have to prove when $H$ is a subgroup of $G$ and the restriction map on the level $H^1$ is injective for arbitray modules, one has $H=G$. can anyone give me some help? is this a classical fact in group cohomology? thanks!