Prove or disprove if $|G|=|K|\neq1$ then the direct product $|G|\times|K|$ will never be a cyclic group

Question

$$\gcd(|G|,|K|) = |G|=|K|\ne1$$ $$|G\times K|=|G||K|=|G|^{2}=|K|^{2}$$ $$G\times K =\{(g,k):g\in G,k\in K\}$$ I need to find if $$\:\exists\:(g,k)^{|G|^2}\in G\times K$$ I can't use: if G and H are cyclic groups whose orders are relatively prime, then G × H is cyclic as well.
Thanks.

Answer 1

Consider the group $H = G \times K$. Then if $\mid G \mid = \mid K \mid = n \not = 1$, the group $H$ has order $n^2$. But it's fairly easy to notice that each element of $H$ has an order dividing $n$. Therefore as no element has the same order as the group, the group can't be cyclic.