{{unsourced, date=September 2022 In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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 ide ...

''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 structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

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

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

, such 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 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 G i ...

of finite

index Index (or its plural form 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 a Halo megastru ...

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, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s of finite index is

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

. *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, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...

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''−1' ...

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, its central series is of finite length or its lower central series terminates with . Intuiti ...

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

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s of compact

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

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 can ...

of residually finite groups is residually finite. In particular, all profinite groups 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 In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. ...

''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 logically 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 str ...

by taking as a

basis of open neighbourhoods In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open ...

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

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

is called the

profinite topology 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. ...

on ''G''. A group is residually finite if, and only if, its profinite topology is Hausdorff. A group whose

cyclic subgroup In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In oth ...

s 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 wor ...

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 Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...

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 The A-Group culture was an ancient culture that flourished between the First and Second Cataracts of the Nile in Nubia. It lasted from 3800 BC to 3100 BC. Overview In 1907, the Egyptologist George A. Reisner first discovered artifacts belong ...

. * 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.

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

External links