Show that if $G= G_1 \oplus G_2$, where $G_1$ and $G_2$ are cyclic of orders $m$ and $n$, then $m$ and $n$ are not uniquely determined by $G$.

Question

Show that if $G= G_1 \oplus G_2$, where $G_1$ and $G_2$ are cyclic of orders $m$ and $n$, respectively, then $m$ and $n$ are not uniquely determined by $G$ in general. [Hint: If $m$ and $n$ are relatively prime, show that $G$ is cyclic of order $mn$.]

I have shown the hint, but I don't understand how this leads to the conclusion of the problem. How do I show this? I would greatly appreciate any help.

Answer 1

Hint: (i) Let $m=(2)(3)$ and $n=5$; (ii) Let $m=2$ and $n=(3)(5)$.