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 :)
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.
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)
