# 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.