Necessary and sufficient condition for one cyclic group to be the subset of some other cyclic group.

Question

My professor gave us this problem, wondering if anyone could help me out:

Suppose a is an element of order n in a group G. Find a necessary and sufficient condition for which $\langle a^r\rangle \subseteq \langle a^s\rangle$. Prove your assertion.

Thanks.

Answer 1

Hint: Without loss of generality we may assume that our group is the integers $0$ to $n-1$ under additon modulo $n$.

Let $d=\gcd(r,n)$ and $e=\gcd(s,n)$. Find a relationship between $d$ and $e$ that is equivalent to the given condition.

Answer 2

Hint: Think about the cyclic group $G = \mathbb{Z} / n\mathbb{Z} = \langle 1 \rangle$ (under addition) for various natural numbers $n$. How does the (cyclic) subgroups look like? Answer: This look like this: $\langle s\rangle$ for an integer $s$. Now try to write down the elements of $\langle r \rangle$ and $\langle s\rangle$ for various values of $r$ and $s$. For example with $n = 10$: $$ \langle 2\rangle = \{0, 2, 4, 6, 8 \}. $$