State and prove the orbit-stabilizer theorem. Deduce that if $x$ is an element of a finite group $G$ then the order of $x$ divides the order of $G$

Prove Cauchy's theorem, that if $p$ is a prime dividing the order of a finite group $G$ then $G$ contains an element of order $p$.

For which positive integers $n$ does there exist a group of order $n$ in which every element (apart from the identity) has order 2?

Give an example of an infinite group in which every element (apart from the identity) has order $2 .$