Should be used with the (group-theory) tag. Free groups are the free objects in the category of groups and can be classified up to isomorphism by their rank. Thus, we can talk about *the* free group of rank $n$, denoted $F_n$.

Free groups are the free objects in the category of groups. This means that if $S$ is some set such that there exists a function $f: S\rightarrow G$ where $G$ is some group then there exists some group homomorphism $\varphi: F_S\rightarrow G$ such that the following diagram commutes,

The universality of free groups implies the set $S$ which they are generated by is important, and indeed one can view a free group over the set $S$ as the set of all words over $S^{\pm 1}$ under the operation of concatenation. This leads to the theory of group presentations.

Free groups can be classified up to isomorphism by their rank. Thus, we can talk about *the* free group of rank $n$, denoted $F_n$.

The standard (classical) reference for free groups is the book "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations" by Wilhelm Magnus, Abraham Karrass and Donald Solitar.

*Note: diagram taken from Wikipedia.*