Residually finite group

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 ${\displaystyle h(g)\neq 1.\,}$^[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 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 ${\displaystyle G}$ is residually finite if, for every ${\displaystyle 1_{G}\neq g\in G}$^[a], there exists some finite group ${\displaystyle F}$ and some group homomorphism ${\displaystyle \phi :G\to F}$ such that ${\displaystyle \phi (g)\neq 1_{F}}$.^[2] There are other equivalent characterizations of residually finite groups:

• A group ${\displaystyle G}$ is residually finite if and only if, for every ${\displaystyle g,h\in G}$ where ${\displaystyle g\neq h}$, there exists some finite group ${\displaystyle F}$ and some group homomorphism ${\displaystyle \phi :G\to F}$ such that ${\displaystyle \phi (g)\neq \phi (h)}$.^[3]
• 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.^[4]
• 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.^[5]

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.^[6]

The integers are an example of an infinite residually finite group. Given any non-zero integer ${\displaystyle k}$ and letting ${\displaystyle m}$ be an integer with ${\displaystyle m>|k|}$, the canonical homomorphism from the integers to the group of integers modulo ${\displaystyle m}$, ${\displaystyle \phi$ :\mathbb {Z} \to \mathbb {Z} /m\mathbb {Z} } , does not map ${\displaystyle k}$ onto ${\displaystyle 0}$.^[7] a similar technique done on the entires of the matrices of ${\displaystyle GL_{n}(\mathbb {Z} )}$, for ${\displaystyle n\geq 1}$, shows that this group is also residually finite.^[8] More generally, all finitely generated abelian groups are residually finite.^[9] Furthermore, the automorphism group of any finitely generated residually finite group will be residually finite.^[10]

Subgroups of a residually finite group are themselves residually finite.^[11] The direct product^[12] and direct sum^[13] of residually finite groups will also be residually finite.

Any inverse limit of residually finite groups is residually finite.^[14] In particular, all profinite groups are residually finite.^[15] One example of a profinite group is the p-adic integers.^[16]

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.^[17]

More examples of groups that are residually finite are free groups^[18], finitely generated nilpotent groups, Polycyclic groups^[19], finitely generated linear groups^[20], and fundamental groups of compact 3-manifolds.^{[citation needed]}

Nonexamples

A divisible group is a group ${\displaystyle G}$ where every ${\displaystyle g\in G}$ and integer ${\displaystyle n\geq 1}$ has an element ${\displaystyle h\in G}$ where ${\displaystyle h^{n}=g}$. 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.^[21] Every nontrivial divisible group fails to be residually finite as every homomorphism from a divisible group to a finite group is trivial.^[22]

Examples of non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are Hopfian groups^[23]. For example the Baumslag–Solitar group ${\displaystyle B(1,m)}$ for ${\displaystyle |m|\geq 2}$ is finitely generated, residually finite, and Hopfian^[24] , but ${\displaystyle B(2,3)}$ is finitely generated yet not Hopfian, and therefore not residually finite.^[25]

Every infinite simple group is not residually finite because the only normal subgroup with a finite index will be the group itself.^[26] 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 signiture 1 is an infinite simple group.^[27] This can be used to show that the subgroup of permutation on the integers generated by the translation ${\displaystyle n\mapsto n+1}$ and the transposition of ${\displaystyle 0}$ and ${\displaystyle 1}$ is a finitely generated Hopfian group that is not residually finite.^[28]

Group extensions of residually finite groups also need not be residually finite.^[29] One counter example is the wreath product of ${\displaystyle A_{5}}$, the alternating group of degree 5, with the integers, both of which are residually finite.^[30]

Properties

Finitely generated residually finite groups have a solvable word problem^[31], meaning there is a procedure where, given the group's generators, one can find the words that equate to the identity element^[32].

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^[33]. A group is residually finite if, and only if, its profinite topology is Hausdorff.^[34] 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.^[31]

A group whose cyclic subgroups are closed in the profinite topology is said to be ${\displaystyle \Pi _{C}\,}$. 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.^{[citation needed]}

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.^{[citation needed]}
• 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.^{[citation needed]}

See also

• Residual property (mathematics)

Notes

1. where ${\displaystyle 1_{G}}$ denotes the identity element of the group ${\displaystyle G}$