1. ## cyclic groups

If n is a positive integer, then $\sigma(n)$ will denote the cyclic group of order n.

If $gcd(m,n)=1$, prove that $\sigma(mn)\cong\sigma(m)\times\sigma(n)$.

2. Originally Posted by deniselim17
If n is a positive integer, then $\sigma(n)$ will denote the cyclic group of order n.

If $gcd(m,n)=1$, prove that $\sigma(mn)\cong\sigma(m)\times\sigma(n)$.
There is a known result which says if $C$ is cyclic group of order $n$ then $C$ is isomorphic to $\mathbb{Z}_n$ (or in your notation $\sigma (n)$). To show that $\mathbb{Z}_m \times \mathbb{Z}_n$ is isomorphic to $\mathbb{Z}_{nm}$ it is sufficient to show $| \mathbb{Z}_m \times \mathbb{Z}_m | = nm$ (which is immediate) and also that $\mathbb{Z}_m\times \mathbb{Z}_n$ is cyclic. Thus, we need to find a generator. Show that $([1]_m,[1]_n)$ has order $nm$ and therefore it must generate the whole group. This will show the group is cyclic and complete the proof.