Center of ambivalent group is elementary abelian 2-group

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Statement

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

Related facts

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.