# Definition:Free Group on Set

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

Jump to navigation
Jump to search
## 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**.