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| In the [[mathematics|mathematical]] field of [[group theory]], a [[Group (mathematics)|group]] ''G'' is '''residually finite''' or '''finitely approximable''' if for every nontrivial element ''g'' in ''G'' there is a [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a finite group, such that
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| :<math>h(g) \neq 1.\,</math>
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| There are a number of equivalent definitions:
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| *A group is residually finite if for each non-identity element in the group, there is a [[normal subgroup]] of finite [[index (group theory)|index]] not containing that element.
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| *A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial.
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| *A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.
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| *A group is residually finite if and only if it can be embedded inside the [[direct product of groups|direct product]] of a family of finite groups.
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| == Examples ==
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| Examples of groups that are residually finite are [[finite group]]s, [[free group]]s, [[finitely generated group|finitely generated]] [[nilpotent group]]s, [[polycyclic-by-finite group]]s, [[finitely generated group|finitely generated]] [[General linear group|linear groups]], and [[fundamental group]]s of [[3-manifold]]s.
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| Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any [[inverse limit]] of residually finite groups is residually finite. In particular, all [[profinite group]]s are residually finite.
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| == Profinite topology ==
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| Every group ''G'' may be made into a [[topological group]] by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in ''G''. The resulting [[topology]] is called the [[profinite group|profinite topology]] on ''G''. A group is residually finite if, and only if, its profinite topology is [[Hausdorff space|Hausdorff]].
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| A group whose cyclic subgroups are closed in the profinite topology is said to be <math>\Pi_C\,</math>.
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| Groups, each of whose finitely generated subgroups are closed in the profinite topology are called '''subgroup separable''' (also '''LERF''', for ''locally extended residually finite'').
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| A group in which every [[conjugacy class]] is closed in the profinite topology is called '''conjugacy separable'''.
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| == Varieties of residually finite groups ==
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| One question is: what are the properties of a [[variety (universal algebra)|variety]] all of whose groups are residually finite? Two results about these are:
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| * Any variety comprising only residually finite groups is generated by an [[A-group]].
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| * For any variety comprising only residually finite groups, it contains a finite group such that all members are embedded in a direct product of that finite group.
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| == See also ==
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| * [[Residual property (mathematics)]]
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| ==External links==
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| * [http://www.turpion.org/php/full/infoFT.phtml?journal_id=im&paper_id=807&year_id=1969&volume_id=3&issue_id=4&fpage=867 Article with proof of some of the above statements]
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| [[Category:Infinite group theory]]
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| [[Category:Properties of groups]]
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They contact me Emilia. The preferred pastime for my kids and me is to play baseball and I'm trying to make it a profession. Hiring has been my profession for some time but I've already applied for another one. For a whilst she's been in South Dakota.
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