This theorem is due to Serre:

Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no element of order $p$ and let $H \leq G$ be an open subgroup. Then $cd_p(G) = cd_p(H)$.

Where $cd_p(G)$ stands for the pth cohomological dimension of $G$.

In fact it follows from:

Suppose $G$ is pro$-p$ and torsion free. If $H \leq G$ is open and $cd_p(H)< \infty$ then $cd_p(G)< \infty$.

These results are mentioned in Serre's book Galois Cohomology, but are not motivated by an example as far as I can see.

What are some interesting examples and applications of the above theorems?



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