Center is direct product-closed

Statement
Suppose $$G_i, i \in I$$ is a collection of groups, and $$G$$ is the external direct product of these. Then, the center of $$G$$ is the external direct product of the centers of each $$G_i, i \in I$$, embedded in the natural way.

Related facts

 * Derived subgroup is direct product-closed
 * Upper central series is direct product-closed
 * Lower central series is direct product-closed

Corollaries

 * Center is finite direct power-closed characteristic