I have two questions on Serre's "Galois cohomology", the section on profinite groups.
1) Proposition 1 on p.4 claims that if $K \subset H$ are two closed subgroups of a profinite group $G$, then there is a continuous section $G/H \to G/K$.
I have no intuition why this should be true: this seems to be so false to me, when I think about $\mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$. But this is Serre, so I suppose I must be stupid somewhere.
2) If $H$ is a closed subgroup of $G$, he defines the index of $H$ in $G$ to be the lcm of index of $H/H \cap U$ in $G/U$ as $U$ varies over all open normal subgroups of $G$. I don't see why the old notion of index doesn't work here - why do we need a special notion? Is it solely for the purpose of defining pro $p$-groups and make sense of the notion of Sylow subgroups?
Thanks!