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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 universal property of free groups - from Wikipedia

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 os 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.

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