# Find all possible subgroups of a cyclic group of order $n$.

Let $$G$$ which is a cyclic group with $$n$$ elements. Find:
a) all possible subgroups
b) all elements which generate whole group.

I know that number of subgroups is the number of total divisors of $$n$$. Moreover if $$l|(|G|=n)$$ then exist exactly one subgroup $$H$$ such that $$|H|=l$$.

However I know how to do this task when I know $$n$$.

For example for $$n=6$$ we have divisors: $$1,2,3,6$$ so we have $$4$$ supgroups.

Meanwhile I have a problem when I don't know $$n$$:

Let $$G= \left\{ e,g,g^{2},\dots ,g^{n-1} \right\}$$ then we have: $$\langle e\rangle=\left\{ e \right\}$$ $$\langle g\rangle =\left\{ e,g,g^{2},\dots ,g^{n-1} \right\}$$ But I already have a problem with $$\langle g^2\rangle$$ because firstly I have: $$\langle g^2\rangle =\left\{ e,g^{2},g^{4},\dots ,g^{n-1}\right\}$$ but I don't know if $$n=2k+1$$ and this is whole $$\langle g^2\rangle$$ or $$n=2k$$ and $$g^2$$ is not a generator of any subgroups.

How can I do this task for every $$n$$?

• About b). If $g$ is a generator, than any $g^k$ is a generator iff $(k, n)=1$. As a corollary, if $n$ is prime, than any non-unit element of $G$ is a generator.
– user615081
Oct 19, 2019 at 15:01

If $$\langle g\rangle =G$$, then $$\mid g^k\mid=\frac n{\operatorname {gcd}(n,k)}$$.
For $$b)$$, using this fact we get that whenever $$\operatorname {gcd}(n,k)=1$$, $$\langle g^k\rangle =G$$.