Free group

Definition

Let $S$ be a set. The free group $F_S$ on $S$ is the set of reduced words on $S$. The product of two reduced words is their concatenation, followed by reduction. A group $G$ is free when it is isomorphic 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, 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.