Definition:Free Group on Set

From ProofWiki
Jump to: navigation, search

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