$G_1$ is a group of order $p$ and $G_2$ a group of order $q$, where $p, q$ are primes. Prove that the direct product $G_1\times G_2$ is cyclic.

Question

So there's a similar question posted (twice) before, but I'm not sure if the proof will be the same. Can I use the fact that the product of $G_1$ of order $p$ and $G_2$ of order $q$ is some group $G$ of order $pq$? If this is true, how do I adapt this to prove my original question? Thanks!

Answer 1

Hint: If $g_i$ is a generator for $G_i$, then $(g_1,g_2)$ is a generator for $G_1 \times G_2$.

It is enough to prove that there are $pq$ different powers of $(g_1,g_2)$.

Answer 2

HINT: $$o((g_1,g_2)) = LCM(o(g_1), o(g_2))$$

Use this to prove that there is an element of order $pq$, which is enough to conclude that the group is cyclic. (Of course we need $p \not = q$, as it's been already mentioned in the comments)