Let G be a group of order 25. Prove that G is cyclic or g^5=e for all g in G.
The group is not necessarity cyclic, how about .
If is cyclic then it is clearly true because and .
If is not cyclic then choose so that . Then is a subgroup of . It cannot (subgroup) have order 1 because it is not identity and it cannot be 25 (the full group) because we assumed it is not cyclic. Thus it must be 5 by Lagrange's theorem.