Rational and nilpotent implies 2-group

Statement
Suppose $$G$$ is a group that is both a fact about::rational group and a fact about::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) uses::Ambivalent and nilpotent implies 2-group
 * 2) uses::Rational implies ambivalent

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