Definition:Free Group on Set
Let $X$ be a set.
that can be defined as follows:
Definition 1: by 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: