Residually finite

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

There are a number of equivalent definitions:

Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic-by-finite groups, finitely generated linear groups, and fundamental groups of 3-manifolds.

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 groups are residually finite.

Counterexamples can be given using the fact that all finitely generated residually finite groups are Hopfian groups. For example the Baumslag–Solitar group B(2,3) is not Hopfian, and therefore not residually finite.

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.

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