Cohomology and finite subgroups of profinite groups

We prove two theorems linking the cohomology of a pro-$p$ group $G$ with the conjugacy classes of its finite subgroups.

The number of conjugacy classes of elementary abelian $p$-subgroups of $G$ is finite if and only if the ring $H^{*}(G,\mathbb{Z} /p)$ is finitely generated modulo nilpotent elements.

If the ring $H^{*}(G,\mathbb{Z} /p)$ is finitely generated, then the number of conjugacy classes of finite subgroups of $G$ is finite.