Residually finite group

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]

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 entries 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]}

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].

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]}

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]}

• Residual property (mathematics)