Odd-order and ambivalent implies trivial

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Statement

Suppose G is an Odd-order group (?) (i.e., a finite group of odd order) that is also an Ambivalent group (?): every element is conjugate to its inverse. Then, G is the trivial group.

Facts used

  1. Odd-order implies solvable
  2. Abelianization of ambivalent group is elementary abelian 2-group
  3. Order of quotient group divides order of group

Proof

By fact (1), G is solvable. Hence, if G is nontrivial, its abelianization is nontrivial. By fact (2), the abelianization of G has order a power of 2. This, however, contradicts fact (3), since G has odd order.