Criterion for $G\oplus H$ to be cyclic. Let G and H be finite cyclic groups. Then $G\oplus H$ is cyclic if and only if $\left | H \right |,\left | G \right |$ are rel. prime.
Proof:
Let $\left | G \right |=m$ and $\left | H \right |=n$ so that $\left | G\oplus H \right |=mn$. To prove the first half of the theorem, assume $G\oplus H$ is cyclic. Suppose that $gcd\left ( m,n \right )=d$ and $\left ( g,h \right )$ is a generator of $G\oplus H$.
Here is where I am unable to understand the theorem from which the author is making inference from (and if I may, very frustrating):
Since $\left ( g,h \right )^{\frac{mn}{d}}=\left ( e,e \right )$ ...
The fact that the author made this deduction implies that the order of $\left ( g,h \right )$ is $\frac{mn}{d}$.
I am unable to understand why the order is mn/d
I've tried looking back at the fundamental theorem of cyclic group, and the theorem for $\left \langle a^{k} \right \rangle=\left \langle a^{gcd\left ( n,k \right )} \right \rangle$ for element $a$ in a group G and k of positive integer. But to no avail. I'm weak in the whole gcd manipulation thingy and the author could be doing such manipulation.
Edit: I had posed this question a while back but received no satisfactory replies. As I'm reading through the book again to refine my understanding, I hope to understand this part of the proof.
Please help. Thanks in advance.