Let $x$ be an element of a finite group $G$. What is meant by the order of $x$ ? Prove that the order of $x$ must divide the order of $G$. [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]

If $G$ is a group of order $n$, and $d$ is a divisor of $n$ with $d<n$, is it always true that $G$ must contain an element of order $d$ ? Justify your answer.

Prove that if $m$ and $n$ are coprime then the group $C_{m} \times C_{n}$ is cyclic.

If $m$ and $n$ are not coprime, can it happen that $C_{m} \times C_{n}$ is cyclic?

[Here $C_{n}$ denotes the cyclic group of order $n$.]