Residually finite

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

An equivalent definition is that a group is residually finite if the intersection of all its finite index normal subgroups is trivial.

Examples of groups that are residually finite are finite groups (trivially), 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.