Rational and nilpotent implies 2-group

Statement

Suppose $G$ is a group that is both a Rational group (?) and a Nilpotent group (?). Then, $G$ must be a 2-group, i.e., it is a group in which every element has finite order and the order is a power of 2.

Related facts

• Rational and abelian implies elementary abelian 2-group

Facts used

1. Ambivalent and nilpotent implies 2-group
2. Rational implies ambivalent

Proof

The proof follows directly from Facts (1) and (2).