Residually finite group

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In the mathematical field of group theory, a group is residually finite or finitely approximable if for every nontrivial element g there is a homomorphism h to a finite group, so that $h(g) \neq 1$ .

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. This 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.

[edit] 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.