Showing that a direct product is non-cyclic

Question

We have that $Z_{2} \times Z_{2}$ is non cyclic this can be easy seen by that $(2,2) \neq 1$ or simply by writing out the table, but I am searching for another method which I was introduced in during class. If I remember correctly it had something to do with LCM and perhaps Lagrange? Does anyone know about this method?

So my question is basically how do I know that the direct product is non-cyclic without using coprime method and by inspection, but using LCM

Answer 1

If $\gcd(m,n)>1$ then $C_m \times C_n$ is not cyclic.

Indeed, let $L=\operatorname{lcm}(m,n)$. Then $g^L = 1$ for all $g \in C_m \times C_n$. Since $$ L = \frac{mn}{\gcd(m,n)} < mn $$ there is no element of order $mn$ in $C_m \times C_n$.