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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 element ''g'' that is not the identity in ''G''  there is a [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a [[finite group]], such that

:<math>h(g) \neq 1.\,</math><ref>{{Cite journal |last=Magnus |first=Wilhelm |date=March 1969 |title=Residually finite groups |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-2/Residually-finite-groups/bams/1183530287.full |journal=Bulletin of the American Mathematical Society |volume=75 |issue=2 |pages=305–316 |doi=10.1090/S0002-9904-1969-12149-X |issn=0002-9904|doi-access=free }}</ref>

There are a number of equivalent definitions:
*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.
*A group is residually finite if and only if the intersection of all its [[subgroup]]s of finite index is [[trivial group|trivial]].  
*A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.
*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.

== Examples ==

Examples of groups that are residually finite are [[finite group]]s, [[free group]]s, [[finitely generated group|finitely generated]] [[nilpotent group]]s, [[Polycyclic group|polycyclic-by-finite group]]s, [[finitely generated group|finitely generated]] [[General linear group|linear groups]], and [[fundamental group]]s of compact [[3-manifold]]s.

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.

Examples of non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are [[Hopfian group]]s. For example the [[Baumslag–Solitar group]] ''B''(2,3) is not Hopfian, and therefore not residually finite.

== Profinite topology ==

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]].

A group whose [[cyclic subgroup]]s are closed in the profinite topology is said to be <math>\Pi_C\,</math>.
Groups each of whose finitely generated subgroups are closed in the profinite topology are called '''subgroup separable''' (also '''LERF''', for ''locally extended residually finite'').
A group in which every [[conjugacy class]] is closed in the profinite topology is called '''conjugacy separable'''.

== Varieties of residually finite groups ==

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:

* Any variety comprising only residually finite groups is generated by an [[A-group]].
* 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.

== See also ==
* [[Residual property (mathematics)]]

==References==
<references />

==External links==
* [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]

[[Category:Infinite group theory]]
[[Category:Properties of groups]]