What is the relation between these two subgroups of a finite cyclic group?

Question

Let $G$ be a finite cyclic group of order $n$ generated by $a$. If $k$, $m$ are integers such that gcd($n, m$) = gcd($n,k$), then what is the relationship between the subgroups generated by $a^k$ and $a^m$?

Answer 1

Let $d := $ gcd($m,n$) $=$ gcd($k,n$). Then $k = k^\prime d$ and $m = m^\prime d$, where $k^\prime$, $m^\prime$ are integers.

Thus $a^k = (a^d)^{k^\prime}$ and $a^m = (a^d)^{m^\prime}$. So $$\langle a^k \rangle \subset \langle a^d \rangle$$ and $$\langle a^m \rangle \subset \langle a^d \rangle.$$

Also, we can write $d = k_0 k + n_1 n = m_0 m + n_2 n$, where $k_0$, $m_0$, $n_1$, $n_2$ are all integers.

We know that $a^n = e$. So $$ a^d = a^{k_0 k + n_1 n} = (a^k)^{k_0},$$ and $$ a^d = a^{m_0 m + n_2 n} = (a^m)^{m_0}.$$ So we can conclude that $$ \langle a^d \rangle \subset \langle a^k \rangle$$ and $$ \langle a^d \rangle \subset \langle a^m \rangle. $$.

Hence $$ \langle a^k \rangle = \langle a^m \rangle = \langle a^d \rangle, $$ as required.