Profinite Group - Properties and Facts

Properties and Facts

• Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is exact on the category of profinite groups. Further, being profinite is an extension property.

• Every closed subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology. If N is a closed normal subgroup of a profinite group G, then the factor group G/N is profinite; the topology arising from the profiniteness agrees with the quotient topology.
• Since every profinite group G is compact Hausdorff, we have a Haar measure on G, which allows us to measure the "size" of subsets of G, compute certain probabilities, and integrate functions on G.
• A subgroup of a profinite group is open if and only if it is closed and has finite index.
• According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group (that is, a profinite group that has a dense finitely-generated subgroup) the subgroups of finite index are open. This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely-generated pro-p groups. The proof uses the classification of finite simple groups.
• As an easy corollary of the Nikolov-Segal result above, any surjective discrete group homomorphism φ: G → H between profinite groups G and H is continuous as long as G is topologically finitely-generated. Indeed, any open subgroup of H is of finite index, so its preimage in G is also of finite index, hence it must be open.
• Suppose G and H are topologically finitely-generated profinite groups which are isomorphic as discrete groups by an isomorphism ι. Then ι is bijective and continuous by the above result. Furthermore, ι−1 is also continuous, so ι is a homeomorphism. Therefore the topology on a topologically finitely-generated profinite group is uniquely determined by its algebraic structure.