Odd-order and ambivalent implies trivial
- Odd-order implies solvable
- Abelianization of ambivalent group is elementary abelian 2-group
- Order of quotient group divides order of group
By fact (1), is solvable. Hence, if is nontrivial, its abelianization is nontrivial. By fact (2), the abelianization of has order a power of . This, however, contradicts fact (3), since has odd order.