1. The problem statement, all variables and given/known data Suppose a cyclic group, G, has only three distinct subgroups: e, G itself, and a subgroup of order 5. What is |G|? What if you replace 5 by p where p is prime? 2. Relevant equations 3. The attempt at a solution So, G has three distinct subgroups. By Lagrange's theorem, the order of the subgroup has to divide the order of the group. So the order of G is a multiple of 5. If we let |G| = 5, then there are only two subgroups, G and e. So we try |G| = 10. Why couldn't 10 be correct? Am I neglecting the fact that G is cyclic? (I know that the answer is actually 25, but am not sure why).