Definition:Free Group on Set/Definition 1

Definition
Let $X$ be a set.

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

Also see

 * Equivalence of Definitions of Free Group on Set