# 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 $(F, \iota)$ where:

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