Define by .

Since is finite it is sufficient to prove that is injective for to be a bijection.

If

And .

Say has order .3. Le G be a group of order where p is prime. Prove that the center G cannot have order .

Pick and construct .

Now prove that is a subgroup of .

Notice that isproperlycontained in .

This forces because by Lagrange's theorem.

But then it must mean that commutes with all of .

Therefore, . A contradiction!