Odd-order and ambivalent implies trivial
Suppose has odd order, is ambivalent, and is nontrivial. Then, there exists a non-identity element in . By fact (1), has odd order, so .
Since is ambivalent, there exists such that . Then, , so and commute. Again by fact (1), has odd order, so . Since commutes with , it must commute with all elements in , and hence with . Thus, . This forces , a contradiction.