Definition:Free Group on Set
(Redirected from Definition:Universal Property of 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:
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.