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

• free groups

• finitely generated$\;$nilpotent groups

• finitely generated$\;$linear group

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 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]