Definition:Free Group on Set

Definition
Let $X$ be a set.

A free group on $X$ is a certain $X$-pointed group, that is, a pair $(F, \iota)$ where:
 * $F$ is a group
 * $\iota : X \to F$ is a mapping

that can be defined as follows:

Definition 1: by universal property
A free group on $X$ is an $X$-pointed group $(F, \iota)$ that satisfies the following universal property:
 * For every $X$-pointed group $(G, \kappa)$ there exists a unique group homomorphism $\phi : F \to G$ such that $\phi \circ \iota = \kappa$, that is, a morphism of pointed groups $F \to G$.

Definition 2: As the group of reduced group words
The free group on $X$ is the pair $(F, \iota)$ where:
 * $F$ is the group of reduced group words on $X$
 * $\iota : X \to F$ is the canonical injection

Also see

 * Equivalence of Definitions of Free Group on Set
 * Definition:Free Abelian Group on Set
 * Definition:Free Monoid on Set