Abelian and ambivalent iff elementary abelian 2-group

Statement

The following are equivalent for a group $G$:

1. $G$ is both an abelian group and an ambivalent group: every element is conjugate to its inverse.
2. $G$ is an elementary abelian 2-group, i.e., it has exponent one or two (this is equivalent to being elementary abelian because exponent two implies abelian).

Related facts

Applications

• Abelianization of ambivalent group is elementary abelian 2-group
• Center of ambivalent group is elementary abelian 2-group
• Odd-order and ambivalent implies trivial

Proof

Abelian and ambivalent to elementary abelian

This follows quite directly: ambivalent 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.

Elementary abelian to abelian and ambivalent

This direction is also immediate.