Suppose a and b are in a group, a has odd order, and . .
In the relation , take the inverse of both sides, to get . It follows that . By the same calculation, , and so on. In fact, is equal to if k is odd, and if k is even.
Now take k to be the order of a, which is odd. Then (the identity element of the group), and . But since k is odd, this is also equal to . So and therefore .