Center of ambivalent group is elementary abelian 2-group

Statement

Suppose $G$ is an ambivalent group. Then, the center of $G$ is an Elementary abelian 2-group (?).

Related facts

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

Facts used

1. Conjugacy-closed subgroup of ambivalent group is ambivalent: A conjugacy-closed subgroup is a subgroup with the property that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.
2. Abelian and ambivalent iff elementary abelian 2-group

Proof

Direct proof

Given: An ambivalent group $G$ with center $Z$.

To prove: For any element $g\in Z$, $g=g^{-1}$ (this suffices because the group is already abelian on account of being the center).

Proof: By the definition of ambivalence, we know that $g$ and $g^{-1}$ are conjugate in $G$. However, $g\in Z$, so this forces that $g$ equals all its conjugates, forcing $g=g^{-1}$.

Fancy proof

The proof follows directly from facts (1) and (2), and the fact that the center of a group is a conjugacy-closed subgroup.