Free product of nontrivial groups is infinite

Statement

Suppose $G$ and $H$ are nontrivial groups (i.e., neither of them is a trivial group). Then, the External free product (?) $G*H$ is infinite.

Related facts

• Free product of nontrivial groups is centerless, which yields that central implies amalgam-characteristic
• Free product of nontrivial groups has no nontrivial finite normal subgroup, which yields that finite normal implies amalgam-characteristic
• Free product of nontrivial groups has no nontrivial periodic normal subgroup, which yields that periodic normal implies amalgam-characteristic