---
tag: 020Z5SEN27
kind: definition
name: Free group
msc: ["20E05"]
defines: 0KTW26C80J
uses: [0FPXK5CHQ3, 085PND4KNT]
sources:
  - url: https://en.wikipedia.org/wiki/Free_group
    locator: Construction
  - url: https://en.wikipedia.org/wiki/Free_group
    locator: lead section
---

## Definition

Let $S$ be a set.
The free group $F_S$ on $S$ is the set of [[reduced words|0KBTP8Q2FS]] on $S$.
The product of two reduced words is their concatenation, followed by reduction.
A [[group|G5S1M9]] $G$ is free when it is [[isomorphic|08MG7H8JW5]] to $F_S$ for some set $S$.

## Remarks

The identity of $F_S$ is the empty word, and the inverse of $t_1 \cdots t_n$ is $t_n^{-1} \cdots t_1^{-1}$.
The set $S$ is a basis of $F_S$. Also $F_S \cong F_T$ if and only if $S$ and $T$ have the same cardinality, which is the rank.
The free group of rank $1$ is the [[infinite cyclic group|02B1T50JD1]], and for $|S| \geq 2$ the centre of $F_S$ is trivial.
A nontrivial finite group is never free, because each element of a basis has infinite order.

## Explanations

### L1 human

A free group has generators but no relations that you do not need.
Two words give the same element only when the group axioms force it, as in $s t = s u u^{-1} t$.
Each group is a quotient of a free group, so free groups are the starting point for presentations.