Free Product of Groups

The free product of two Groups $G,H$ is the quotient of the Free Group on the set $G+H$ by the Equivalence Relation generated by

• $x,g_{1},g_{2},y\sim x,g_{1}g_{2},y$

• $x,h_{1},h_{2},y\sim x,h_{1}h_{2},y$

Properties

• Somewhat confusingly, the free product of groups is the Coproduct in the Category of Groups.