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