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

The symmetric group Sym(n)Sym(n) may be generated from

References

Textbooks:

See also

Last revised on April 17, 2021 at 14:14:47. See the history of this page for a list of all contributions to it.