What is the order of a coset?

Question

I'm trying to read Cauchy's Theorem on wikipedia (The abelian part of Proof 1) and they say that $G/H$ contains an element of order $p$. What does the order of a right coset mean? Is it the number of elements in the right coset?

Answer 1

For example, consider $G:=\Bbb Q/\Bbb Z$. It has an element of order $3$. It means we produce an element $r+\Bbb Z$ such that $3(r+\Bbb Z)=(r+\Bbb Z)+(r+\Bbb Z)+(r+\Bbb Z)=3r+\Bbb Z=0+\Bbb Z$. Such an element here is $\frac{1}{3}+\Bbb Z$

$\frac{1}{3}+\Bbb Z$ is a single element of $\Bbb Q/\Bbb Z$ here, eventhough it is a coset, because $\Bbb Q/\Bbb Z$ is itself a group.