# Odd-order and ambivalent implies trivial

## Statement

Suppose 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, is the trivial group.

## Facts used

## Proof

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.