If is a group of odd order, prove that for any nonidentity element , and are not conjugate.

Suppose to the contrary that for some . Note then that and that .

We now prove a few facts.

Fact 1: If is odd, then . Proof: If , then clearly . Suppose that the conclusion holds for some odd . Then . By induction, the conclusion holds for all odd positive .

Fact 2: If is even, then . Proof: If , then clearly . Suppose that the conclusion holds for some even . Then . By induction, the conclusion holds for all even nonnegative . .

Fact 3: If is odd, then . Proof: If , we have . Now suppose the conclusion holds for some odd ; we have . By induction, the conclusion holds for all odd positive .

Fact 4: If is even, then . Proof: If , we have . Now suppose the conclusion holds for some even . Then . By induction, the conclusion holds for all even nonnegative .

Suppose ; now is finite and moreover odd since is odd.

Note that . On the other hand, . Thus , a contradiction.

Hence and are never conjugate in a finite group of odd order.

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A typo in fact3, m=1.

Also some links of other solutions:

http://math.berkeley.edu/~daffyd/113/hw4sol.pdf

http://www.mymathforum.com/problems/GT/GT7.pdf