Rational and nilpotent implies 2-group

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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

Facts used

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

Proof

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