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
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There are a number of equivalent definitions:
Examples
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, so for example the Baumslag–Solitar group B(2,3) is not Hopfian.
Profinite topology
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
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: