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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!

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    @Pete But about Serre's publications, particularly his books, even if small mistakes are inevitable, let me report what he said once about his method, which he calls " n = n + 1". He writes a first draft, and keeps it in his drawer for a while. Some time later, he writes a second draft, he compares the two, then puts both in his drawer. And so on, until n = n + 1 essentially.2016-06-05

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As for 1): the section is taking place in the category of topological spaces, not groups. (In fact, since $K$ and $H$ are not assumed to be normal, $G/H$ and $G/K$ will in general not have a group structure to be preserved.) In the case of discrete groups this is just the easy (with suitable set-theoretic goodwill...) fact that for every surjective function $f: X \rightarrow Y$ there is a map $\iota: Y \rightarrow X$ such that $f \circ \iota$ is the identity function on $Y$.

As for 2): what "old notion" do you mean exactly: the cardinality of the coset space? The point is that this will in general be infinite. We want the index to be a supernatural number -- essentially, a purely formal limit of natural numbers -- rather than an infinite cardinal number. And yes, this is probably most important for Sylow theory...certainly that's how Serre uses it in his book.

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    Yes I mean cardinality of coset space. Thanks!2012-03-18