Rational and abelian implies elementary abelian 2-group

Statement

Suppose $G$ is a group that is both a rational group and an abelian group. Then, $G$ must be an elementary abelian 2-group, i.e., an elementary abelian 2-group, i.e., it is isomorphic to the additive group of a vector space over field:F2, or equivalently, it is an abelian group of exponent dividing $2$.

Related facts

• Rational and nilpotent implies 2-group
• Ambivalent and abelian implies elementary abelian 2-group
• Center of rational group is elementary abelian 2-group
• Center of ambivalent group is elementary abelian 2-group
• Ambivalent and nilpotent implies 2-group

Proof

This follows quite directly: rational implies that every element is conjugate to its inverse, which in the abelian case forces every element to be equal to its inverse, thus forcing all elements to have order dividing 2.