Residually finite group


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 a normal 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 the direct product of a family of finite groups.

Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups and polycyclic-by-finite groups.

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 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 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 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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