Understanding a Theorem regarding Order of elements in a cyclic group This is part of practice midterm that I have been given (our prof doesn't post any solutions to it) I'd like to know whats right before I write the midterm on Monday this was actually a 4 part question, I'm posting just 1 piece as a question in its own cause it was much too long.
Let $G$ be an abelian group.
Suppose that $a$ is in $G$ and has order $m$ (such that $m$ is finite) and that the positive integer $k$ divides $m$.
(ii) Let $a$ be in $G$ and $l$ be a positive integer. State the theorem which says $\langle a^l\rangle=\langle a^k\rangle$ for some $k$ which divides $m$.
I have a theorem in my textbook that says as follows.
Let G be a finite cyclic group of order n with $a \in G$ as a generator for any integer m, the subgroup generated by $a^{m}$ is the same as the subgroup generated by $ a^{d}$ where $d=(m,n)$.
if this is the correct theorem can someone please show me why? thanks

Yes you can apply the theorem you cite immediately with $(\langle a\rangle,m,l)$ repectively in the roles of $(G,n,m)$ (here $\langle a\rangle$ is the cyclic subgroup generated by $a$); the desired value for $k$ is the value $d$ of the theorem, which is, in view of the name changes, $k=\gcd(m,l)$.
