## Exam failed, could anyone solve those? (Galois cohomology 2 tasks) [closed]

Could someone please try to solve those two tasks, i'm trying to solve them for over a week and i can't do it. i will be gratefull for quick answers :)

1. Let $\pi : G_{1} \rightarrow G_{2}$ be a continuous epimorphism of profinite groups $G_{1}$ and $G_{2}$. Let $A$ be a discreet $G_{2}$ module. On $A$ in a natural way we place the $G_{1}$ structure through $\pi$ homomorphism. Show that the natural homomorphism $H^{1}(G_{2}, A) → H^{1}(G_{1}, A)$ is injective.

2. Let $f: G_{1} \rightarrow G_{2}$ be a continuous homomorphism of profinite groups $G_{1}$ and $G_{2}$. Show that:

• for every profinite subgroup $H_{1} \subset G_{1}$ image $f(H_{1}) \subset G_{2}$ is a profinite subgroup,
• for every profinite subgroup $H_{2} \subset G_{2}$ preimage $f^{-1}(H_{2}) \subset G_{1}$ is a profinite subgroup.

BTW. they're not homework but exam questions (i really had/have no idea how to solve them)

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This breaks my heart,but I'd be irresponsible if I didn't vote to close. – Andrew L Dec 29 2010 at 15:20
easier than homework? not in Poland... and the Professor won't tell me how to do them because they might show up again on 2nd exam, or something similiar. – unknown (google) Dec 29 2010 at 15:57
@Andrew L : You don't have the ability to vote to close, which requires 3000 rep. If you ever get to that point, then you won't vote to close by leaving a comment, but there will be a "close" button at the bottom of the question. Until then, please refrain from trying to impersonate someone with the ability to vote to close. – Andy Putman Dec 29 2010 at 16:04
Dear Unknown, These questions are straightforward if you know the basic theory of group cohomology and profinite groups. If you really have no idea how to solve them, you should learn these theories first, rather than just asking for solutions on the internet. – Emerton Dec 29 2010 at 17:32
@Emerton: you are no fun! – Igor Rivin Dec 29 2010 at 20:52