nLab finitely generated group

Contents

Contents

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)Sym(n) of permutations of nn \in \mathbb{N} elements may be generated from:

Example

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

References

Textbooks:

See also

Discussion in homotopy type theory/univalent foundations of mathematics:

Last revised on July 16, 2022 at 13:29:41. See the history of this page for a list of all contributions to it.