In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

''h'' from ''G'' to a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

, such that :

h(g) \neq 1.\,

{{Cite journal , last=Magnus , first=Wilhelm , date=March 1969 , title=Residually finite groups , url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-2/Residually-finite-groups/bams/1183530287.full , journal=Bulletin of the American Mathematical Society , volume=75 , issue=2 , pages=305–316 , doi=10.1090/S0002-9904-1969-12149-X , issn=0002-9904, doi-access=free 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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

of finite

index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

not containing that element. *A group is residually finite if and only if the intersection of all its

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

s 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 In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

of a family of finite groups.

Examples

Examples of groups that are residually finite are

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

s, finitely generated

nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . I ...

s, polycyclic-by-finite groups, finitely generated linear groups, and

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

s of compact

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

s. Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any

inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...

of residually finite groups is residually finite. In particular, all

profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...

s are residually finite. Examples of non-residually finite groups can be constructed 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.

Profinite topology

Every group ''G'' may be made into a

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

by taking as a

basis of open neighbourhoods Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...

of the identity, the collection of all normal subgroups of finite index in ''G''. The resulting

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

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 In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...

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.

References

External links

Article with proof of some of the above statements
Infinite group theory Properties of groups

Examples

Profinite topology

Varieties of residually finite groups

See also

References

External links