Free group

In mathematics, the free group ${\displaystyle F_{S}}$ over a given set ${\displaystyle S}$ consists of all words that can be built from members of ${\displaystyle S}$, considering two words to be different unless their equality follows from the group axioms (e.g. ${\displaystyle st=suu^{-1}t}$ but ${\displaystyle s\neq t^{-1}}$ for ${\displaystyle s,t,u\in S}$). The members of ${\displaystyle S}$ are called generators of ${\displaystyle F_{S}}$, and the number of generators is the rank of the free group. An arbitrary group ${\displaystyle G}$ is called free if it is isomorphic to ${\displaystyle F_{S}}$ for some subset ${\displaystyle S}$ of ${\displaystyle G}$, that is, if there is a subset ${\displaystyle S}$ of ${\displaystyle G}$ such that every element of ${\displaystyle G}$ can be written in exactly one way as a product of finitely many elements of ${\displaystyle S}$ and their inverses (disregarding trivial variations such as ${\displaystyle st=suu^{-1}t}$).

A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.