nLab finitely generated group

Idea

A finitely generated group is a (discrete) group (equipped) with a finite set of generators, hence of elements such that any other element can be written as a product of these.

Properties

Relation to Cayley graphs

Associated with a set of generators on a group is the corresponding Cayley graph, whose vertices are the group elements which are connected by an edge if they differ by (left, say) multiplication with one of the generators.

The corresponding graph distance equips the group with a metric (the word metric) and thus makes it an object of geometric group theory.

Examples

Example

(symmetric group)
The symmetric group $Sym(n)$ of permutations of $n \in \mathbb{N}$ elements may be generated from:

• all transposition permutations – the corresponding Cayley graph distance is the original Cayley distance;

• the adjacent transpositions – the corresponding Cayley graph distance is known as the Kendall tau distance.

Example

(braid group)
The braid group $Br(n)$ on $n \in \mathrm{N}$ strands is finitely generated via the Artin presentation.

Related concepts

• finite group, profinite group

• generators and relations

• finitely generated object

  • finitely generated algebra

  • finitely generated group

  • finitely generated module, finitely presented module

• Solovay-Kitaev theorem

References

Textbooks:

• Cornelia Druţu, Michael Kapovich, Chapter 7 of: Geometric group theory, Colloquium Publications 63, AMS 2018 (ISBN:978-1-4704-1104-6, pdf)

See also

• Wikipedia, Finitely generated group

Discussion in homotopy type theory/univalent foundations of mathematics:

• Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 6 of: Symmetry (2021) $[$pdf$]$