It's that easy.If G is a group of order 25, prove that either G is cyclic or else every nonidentity element of G has order 5.
Could you just say:
Let G be a group of order 25. If G is cyclic, then we are done.
Suppose G is not cyclic.
All elements of G have order that divides 25, so the order of any element is 1, 5, or 25. If G is not cyclic, then it has no element of order 25.
Then, since we are talking about nonidentity elements, they can not have order 1. Hence all nonidentity elements have order 5.
Is it that easy or am I missing something?