Sylow's theorem in profinite groups

Name
The result is named after a similar and very fundamental result called Sylow's theorem that applies to finite groups.

Statement
Suppose $$G$$ is a profinite group and $$p$$ is a prime number. A $$p$$-Sylow subgroup of $$G$$ is a closed subgroup $$P$$ of $$G$$ such that the order (in the sense of order of a profinite group) is a power of $$p$$ and its index (in the sense of index of a closed subgroup in a profinite group) is relatively prime to $$p$$. Then:


 * Existence: There exist $$p$$-Sylow subgroups of $$G$$.
 * Conjugacy: If $$P, Q$$ are two $$p$$-Sylow subgroups of $$G$$, then there exists $$g \in G$$ such that $$gPg^{-1} = Q$$.
 * Domination: If $$P$$ is a $$p$$-Sylow subgroup of $$G$$ and $$Q$$ is a pro-$$p$$-subgroup of $$G$$, then there exists $$g \in G$$ such that $$gQg^{-1} \le P$$.

There is no equivalent of the congruence condition for the Sylow's theorem in finite groups.