# Center of ambivalent group is elementary abelian 2-group

From Groupprops

## Statement

Suppose is an ambivalent group. Then, the center of 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

- 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.
- Abelian and ambivalent iff elementary abelian 2-group

## Proof

### Direct proof

**Given**: An ambivalent group with center .

**To prove**: For any element , (this suffices because the group is already abelian on account of being the center).

**Proof**: By the definition of ambivalence, we know that and are conjugate in . However, , so this forces that equals all its conjugates, forcing .

### 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.