Ambivalent and nilpotent implies 2-group

Statement

Suppose $G$ is a group that is both a ambivalent group and a nilpotent group. Then, $G$ is a 2-group, i.e., the order of every element of $G$ is a power of 2. In fact, $G$ is a group of finite exponent and the log of the exponent to base 2 is at most the nilpotency class of $G$.

Applications

• Rational and nilpotent implies 2-group

Facts used

1. Center of ambivalent group is elementary abelian 2-group
2. Ambivalence is quotient-closed

Proof

Uses mathematical induction with Facts (1) and (2), considering the center and the quotient by the center.

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