Free product of nontrivial groups is centerless

Statement
Suppose $$G$$ and $$H$$ are nontrivial groups (i.e., neither of them is a trivial group). Then, the fact about::external free product $$G * H$$ is a fact about::centerless group: its center is trivial.

Related facts

 * Free product of nontrivial groups is infinite
 * Central implies amalgam-characteristic
 * Normal subgroup contained in upper central series member is amalgam-characteristic