Sylow's theorem in profinite groups

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


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.