---
tag: 0KTW26C80J
kind: concept
name: Free group
msc: ["20E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Free_group
    locator: lead section
same_as:
  wikidata: Q431078
  wikipedia: Free group
---

## Summary

The free group on a set $S$ is the group of reduced words on $S$, with concatenation followed by reduction as the operation.
It is the group generated by $S$ with no relations other than those that the group axioms force.