Cyclic Group $Z_{n}$
How do I find all the Subgroups of $Z_{200}$ under additive modulo 200? And how many subgroups are there? I know the definition of subgroups but to find all the subgroups of such a large set is a bit difficult for me. Is there a smart way to find it?
The answer to this question is 12 and I not sure if this is correct.
Denote by $d(n)$ the number of positive divisors of $n$. Then $\mathbb{Z}_n$ has $d(n)$ subgroups, because a cyclic group of order $n$ has a unique subgroup for each divisor $d\mid n$, see here:
Subgroups of a cyclic group and their order.
Since $d(200)=12$, we are done.