Profinite Sylow subgroup
= Profinite Sylow subgroup
{title2=$P\le G$}
A Sylow pro-$p$ subgroup of a <profinite group> is a closed pro-$p$ subgroup whose images are Sylow $p$-subgroups in every finite quotient. Such subgroups exist and are conjugate. Every open subgroup containing one has index prime to $p$. They allow prime-to-$p$ finite extensions to detect $p$-primary <Galois cohomology>.