Center of ambivalent group is elementary abelian 2-group

Statement
Suppose $$G$$ is an ambivalent group. Then, the center of $$G$$ is an fact about::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) uses::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) uses::Abelian and ambivalent iff elementary abelian 2-group

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.