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 $g=g^{-1}$.