For a free product $A*B$ of groups $A$ and $B$, there is the exact sequence

$1 \to [A,B] \to A*B \to A \times B \to 1$

where $[A,B]$ is the subgroup generated by all elements $[a,b]=aba^{-1}b^{-1}$ and $A \times B$ is the direct product group. The first map is the inclusion and the second one is the intuitive one. This sequence is important for combinatorial and geometric group theory.