Definition:Free Group on Set

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Definition

Let $X$ be a set.


A free group on $X$ is a certain $X$-pointed group, that is, a pair $\struct {F, \iota}$ where:

$F$ is a group
$\iota : X \to F$ is a mapping

that can be defined as follows:


Definition 1

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 $F \to G$.


Definition 2

The free group on $X$ is the pair $\struct {F, \iota}$ such that:

$F$ is the group of reduced group words on $X$
$\iota : X \to F$ is the canonical injection.


Also see

  • Results about free groups can be found here.