Sylow's theorem in profinite groups
From Groupprops
Name
The result is named after a similar and very fundamental result called Sylow's theorem that applies to finite groups.
Statement
Suppose is a profinite group and is a prime number. A -Sylow subgroup of is a closed subgroup of such that the order (in the sense of order of a profinite group) is a power of and its index (in the sense of index of a closed subgroup in a profinite group) is relatively prime to . Then:
- Existence: There exist -Sylow subgroups of .
- Conjugacy: If are two -Sylow subgroups of , then there exists such that .
- Domination: If is a -Sylow subgroup of and is a pro--subgroup of , then there exists such that .
There is no equivalent of the congruence condition for the Sylow's theorem in finite groups.