nLab residually finite group

Idea

A group is residually finite if it has a good supply of subgroups of finite index.

Definition

Definition

A group $G$ is called residually finite if it satisfies the following equivalent conditions:

1. For each $g \in G$, $g \neq \mathrm{e}$,

   1. there exists a normal subgroup $N \subset G$ of finite index $[G\colon N] \in \mathbb{N}$ such that $g \notin N$.

   2. there exists a finite group $K$ and a group homomorphism $\phi \,\colon\, G \to K$ such that $\phi(g) \neq \mathrm{e}$.

2. The intersection of all subgroups $H \subset G$ of finite index $[G\colon H] \in \mathbb{N}$ is trivial.

3. The intersection of all normal subgroups $N \subset G$ of finite index $[G\colon N] \in \mathbb{N}$ is trivial.

4. $G$ is a subgroup of a direct product of finite groups.

Examples

Example

Classes of examples of residually finite groups include:

• finite groups

• profinite groups

• finitely generated$\;$nilpotent groups

• finitely generated$\;$linear groups.

Example

Free groups are residually finite.

(cf. Hempel 1972, Cohen 1989 pp 7 & 11, Ivanov 2010)

Example

Surface groups (fundamental groups of surfaces) are residually finite.

(Hempel 1972)

Example

The plain braid groups are residually finite.

(Magnus 1969 2.III, Rolfsen 2003 Thm. 2.5)

Example

The mapping class group of a closed oriented surface is residually finite.

(Grossman 1974)

References

Review:

• Wilhelm Magnus: Residually finite groups, Bull. Amer. Math. Soc. 75 2 (1969) 305-316 [euclid:bams/1183530287, ams:1969-75-02/S0002-9904-1969-12149-X]

• Tullio Ceccherini-Silberstein, Michel Coornaert: Residually Finite Groups, chapter 2 of: Cellular Automata and Groups [doi:10.1007/978-3-642-14034-1_2, pdf]

See also:

• Wikipedia, Residually finite group

On residual finiteness of free groups:

• F. Levi: Über die Untergruppen der freien Gruppen [2. Mitteilung], Math. Z. 37 (1933) 90-97 [doi:10.1007/BF01474565]

• Hempel 1972

• Daniel E. Cohen, pp 7,11: Combinatorial group theory: a topological approach, Cambridge University Press (1989) [doi:10.1017/CBO9780511565878]

• Sergei Ivanov, MO comment (2010) [MO:a/20485]

On residual finiteness of fundamental groups of surfaces:

• John Hempel: Residual finiteness of surface groups, Proc. Amer. Math. Soc. 32 1 (1972) 323 [doi:10.1090/S0002-9939-1972-0295352-2]

On residual finiteness of braid groups:

• Dale Rolfsen, Thm. 2.5 in: New developments in the theory of Artin’s braid groups, Topology and its Applications 127 1–2 (2003) 77-90 [doi:10.1016/S0166-8641(02)00054-8, pdf]

On residual finiteness of mapping class groups:

• Edna K. Grossman: On the residual finiteness of certain mapping class groups, J. London Math. Soc. s2-9 1 (1974) 160–164 [doi;10.1112/jlms/s2-9.1.160]