Center of rational group is elementary abelian 2-group

Statement
Suppose $$G$$ is a rational group and $$Z$$ is the center of $$G$$. Then, $$Z$$ is an elementary abelian 2-group. Note that this includes the possibility of $$Z$$ being trivial.

Related facts

 * Rational and abelian implies elementary abelian 2-group
 * Center of ambivalent group is elementary abelian 2-group

Proof
Given: A rational group $$G$$ with center $$Z$$.

To prove: For any element $$g \in Z$$, $$g = g^{-1}$$ (this suffices because the group is already abelian on account of being the center).

Proof: By the definition of rationality, we know that $$g$$ and $$g^{-1}$$ are conjugate in $$G$$. However, $$g \in Z$$, so this forces that $$g$$ equals all its conjugates, forcing $$g = g^{-1}$$.