Since is finite it is sufficient to prove that is injective for to be a bijection.
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 is properly contained in .
This forces because by Lagrange's theorem.
But then it must mean that commutes with all of .
Therefore, . A contradiction!