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

• Results about free groups can be found here.

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