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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?


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Never suppose you're stupid because a great mathematician says something you don't understand ; it is reasonable to assume you don't understand what he says just because you don't understand the theory behind his speech, but even the greatest make mistakes. – Patrick Da Silva Mar 18 '12 at 5:18
@PatrickDaSilva, Thanks. :) – user27126 Mar 18 '12 at 5:22
@Patrick: "[E]ven the greatest make mistakes." That's true, and the proof in my mind is that even Serre makes mistakes (for me, he really is the greatest)....but much more rarely than most of us. Certainly when reading his works the assumption that he's correct and you are somehow misunderstanding something is going to be correct the vast majority of the time. – Pete L. Clark Mar 18 '12 at 5:38
@Pete : Nowhere did I meant to imply that Serre does a ton of mistakes, just that not understanding Serre does not mean you are stupid, nor making mistakes makes you stupid. – Patrick Da Silva Mar 18 '12 at 6:02
@Patrick: And nowhere did I mean to imply that you implied that he did. :) – Pete L. Clark Mar 18 '12 at 6:03
up vote 4 down vote accepted

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! – user27126 Mar 18 '12 at 5:22

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