- Residually finite group
In the mathematical field of

group theory , a group "G" is**residually finite**or**finitely approximable**if for every nontrivial element "g" in "G" there is a homomorphism "h" from "G" to a finite group, such that:$h(g)\; eq\; 1.,$

There are a number of equivalent definitions:

*A group is residually finite if for each non-identity element in the group, there is anormal subgroup of finite index not containing that element.

*A group is residually finite if and only if the intersection of all its subgroups of finite index is 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 thedirect product of a family of finite groups.Examples of groups that are residually finite are

finite group s,free group s, finitely generatednilpotent group s andpolycyclic-by-finite group s.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 resultingtopology is called the**profinite topology**on "G". A group is residually finite if, and only if, its profinite topology is Hausdorff.A group whose cyclic subgroups are closed in the profinite topology is said to be $Pi\_C,$.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 everyconjugacy 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 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.**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*]

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