Residually finite group

Template:Short description Template:More citations needed In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group, such that $h(g)\ne 1.$Template:Sfn

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 of finite index not containing that element.
• A group is residually finite if and only if the intersection of all its subgroups 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 of a family of finite groups.

Definition

A group $G$ is residually finite if, for every $1_G\ne g\in G$Template:Efn, there exists some finite group $F$ and some group homomorphism $\varphi :G\to F$ such that $\varphi (g)\ne 1_F$.Template:Sfn There are other equivalent characterizations of residually finite groups:

• A group $G$ is residually finite if and only if, for every $g,h\in G$ where $g\ne h$, there exists some finite group $F$ and some group homomorphism $\varphi :G\to F$ such that $\varphi (g)\ne \varphi (h)$.Template:Sfn
• A group is residually finite if and only if its residual subgroup (or profinite kernel) is trivial. The residual subgroup of a group is the intersection of all subgroups that have a finite index, or equivalently, the intersection of all normal subgroups of finite index.Template:Sfn
• A group is residually finite if and only if it is isomorphic to a subgroup of a direct product of a family of finite groups.Template:Sfn

Examples

Every finite group is residually finite. This can be shown by considering the group itself as the finite group, and its identity homomorphism as the homomorphism to a finite group.Template:Sfn

The integers are an example of an infinite residually finite group. Given any non-zero integer $k$ and letting $m$ be an integer with $m>|k|$, the canonical homomorphism from the integers to the group of integers modulo $m$, $\varphi :\mathbb{Z}\to \mathbb{Z}/m\mathbb{Z}$, does not map $k$ onto $0$.Template:Sfn a similar technique done on the entries of the matrices of $G{L}_n(\mathbb{Z})$, for $n\ge 1$, shows that this group is also residually finite.Template:Sfn More generally, all finitely generated abelian groups are residually finite.Template:Sfn Furthermore, the automorphism group of any finitely generated residually finite group will be residually finite.Template:Sfn

Subgroups of a residually finite group are themselves residually finite.Template:Sfn The direct productTemplate:Sfn and direct sumTemplate:Sfn of residually finite groups will also be residually finite.

Any inverse limit of residually finite groups is residually finite.Template:Sfn In particular, all profinite groups are residually finite.Template:Sfn One example of a profinite group is the p-adic integers.Template:Sfn

If a group has a subgroup of finite index which is residually finite (that is, a virtually residually finite group), then said group is also residually finite.Template:Sfn

More examples of groups that are residually finite are free groupsTemplate:Sfn, finitely generated nilpotent groups, polycyclic groupsTemplate:Sfn, finitely generated linear groupsTemplate:Sfn, and fundamental groups of compact 3-manifolds.

Nonexamples

A divisible group is a group $G$ where every $g\in G$ and integer $n\ge 1$ has an element $h\in G$ where $h^n=g$. Examples of nontrivial divisible groups include the rational numbers, the real numbers, the complex numbers, the additive group of a vector space over the rationals, and the additive group of every field with characteristic 0.Template:Sfn Every nontrivial divisible group fails to be residually finite as every homomorphism from a divisible group to a finite group is trivial.Template:Sfn

Examples of non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are Hopfian groupsTemplate:Sfn. For example the Baumslag–Solitar group $B(1,m)$ for $|m|\ge 2$ is finitely generated, residually finite, and HopfianTemplate:Sfn , but $B(2,3)$ is finitely generated yet not Hopfian, and therefore not residually finite.Template:Sfn

Every infinite simple group is not residually finite because the only normal subgroup with a finite index will be the group itself.Template:Sfn This implies that the group of permutations on an infinite set with finite support is not residually finite as the subgroup with the permutations of signature 1 is an infinite simple group.Template:Sfn This can be used to show that the subgroup of permutation on the integers generated by the translation $n\mapsto n+1$ and the transposition of $0$ and $1$ is a finitely generated Hopfian group that is not residually finite.Template:Sfn

Group extensions of residually finite groups also need not be residually finite.Template:Sfn One counter example is the wreath product of $A_5$, the alternating group of degree 5, with the integers, both of which are residually finite.Template:Sfn

Properties

Finitely presented residually finite groups have a solvable word problemTemplate:Sfn, meaning there is a procedure where, given the group's generators, one can find the words that equate to the identity elementTemplate:Sfn.

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 GTemplate:Sfn. A group is residually finite if, and only if, its profinite topology is Hausdorff.Template:Sfn If this group is also infinite and finitely generated, then said topology is totally disconnected, and the completion is the inverse limit of a sequence of finite quotients of this group, making it a profinite group.Template:Sfn

A group whose cyclic subgroups are closed in the profinite topology is said to be ${\mathrm{\Pi }}_C$. 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.Template:Cn

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.Template:Cn
• 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.Template:Cn

See also

• Residual property (mathematics)

References

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