I have a question: when I find a quotient group, I actually find the group of all cosets? I mean, every element in a quotient group is actually a coset?

For example, I have a confusing question:

Let (Q,+) be the rational group, (Z,+) be the integer a subgroup of Q.

Prove that every element in Q/Z has a finite order. (Does this mean I need to prove that every coset has a finite number of elements? or, what?)

Anyway, I googled this question, and I found a proof for that:

let z+q be an element in Q/Z, while and ,

then , while and .

Then,

therefore, <<Why is that?! What is the neutral element in Q/Z? I don't understand this>>

Please help me,

Thank you very much