---
tag: TJKANTKASZ
kind: concept
name: Finitely presented group
msc: [20F05]
specializes: [082DWWXMRJ]
sources:
  - url: https://en.wikipedia.org/wiki/Presentation_of_a_group
---

## Summary

A finitely presented group is a group that has a [[presentation|0JGEQXQ4AC]] with finitely many generators and finitely many relations.
Such a group can be described by a finite amount of data, so it can be handled by algorithms and computers.
Finitely presented groups are the natural objects of combinatorial group theory and arise as [[fundamental groups|S1XK3HXHFX]] of compact manifolds and finite complexes.

## Explanations

### L1 human

A finitely presented group is given by a finite list of letters and a finite list of rules that say which words equal the identity.
For example, $\langle a, b \mid a b a^{-1} b^{-1} \rangle$ has two letters and one rule, $ab = ba$, and it is the group $\Z^2$.
Every finite group is finitely presented, since its whole multiplication table is a finite list of rules.
Some finitely generated groups need infinitely many rules, however one chooses the generators.