I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. The cyclic subgroup of G generated by some a is (a) = {$n*a$ for $n$ in $ Z$}. So for example the cyclic subgroup <45> in G is {$0,15,30,45$}. By the same token, the cyclic subgroup <15> in G is {$0,15,30,45$}.
I know there is some relation between gcd(a,|G|) (|G| being the cardinality or order of G), but I can't explain it. Returning to our example, the gcd(45,60) is 15 (which is the reason that they have the same cyclic subgroup in $Z_{60}$, but why is that so?
Would someone be able to give me a very simplified explanation into this? Thank you very much for your help!