Free groups (statement)

Definition. $F_2$ is the set of reduced words in the alphabet $\{a,b,a^{-1},b^{-1}\}$ (no adjacent $x$ and $x^{-1}$), with concatenation followed by reduction as operation.

Structure. $F_2$ is infinite; it has a presentation $F_2=\langle a,b:\emptyset\rangle$ (no relations).

Non-abelian. The words $ab$ and $ba$ are distinct reduced words, so $ab\neq ba$.

Universal property. For any group $G$ and any $g,h\in G$, there is a unique homomorphism $F_2\to G$ sending $a\mapsto g$, $b\mapsto h$.

Conclusion. $F_2$ is the free group on two generators; it is infinite and non-abelian.