# Applications of one of Serre's Theorems

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?