Editing Profinite group (section)

==Properties and facts==

* Every [[direct product of groups|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 functor|exact]] on the category of profinite groups. Further, being profinite is an extension property.
* Every [[closed set|closed]] subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the [[subspace (topology)|subspace topology]]. If <math>N</math> is a closed normal subgroup of a profinite group <math>G,</math> then the [[factor group]] <math>G / N</math> is profinite; the topology arising from the profiniteness agrees with the [[quotient topology]].
* Since every profinite group <math>G</math> is compact Hausdorff, there exists a [[Haar measure]] on <math>G,</math> which allows us to measure the "size" of subsets of <math>G,</math> compute certain [[probabilities]], and [[integral|integrate]] functions on <math>G.</math>
* A subgroup of a profinite group is open if and only if it is closed and has finite [[Index of a subgroup|index]].
* According to a theorem of [[Nikolay Nikolov (mathematician)|Nikolay Nikolov]] and [[Dan Segal]], in any topologically finitely generated profinite group (that is, a profinite group that has a [[dense set|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 group|pro-<math>p</math> group]]s. The proof uses the [[classification of finite simple groups]].
* As an easy corollary of the Nikolov–Segal result above, {{em|any}} surjective discrete group homomorphism <math>\varphi : G \to H</math> between profinite groups <math>G</math> and <math>H</math> is continuous as long as <math>G</math> is topologically finitely generated. Indeed, any open subgroup of <math>H</math> is of finite index, so its preimage in <math>G</math> is also of finite index, and hence it must be open.
* Suppose <math>G</math> and <math>H</math> are topologically finitely generated profinite groups that are isomorphic as discrete groups by an isomorphism <math>\iota.</math> Then <math>\iota</math> is bijective and continuous by the above result. Furthermore, <math>\iota^{-1}</math> is also continuous, so <math>\iota</math> is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its {{em|algebraic}} structure.