One sentence from Amstrong's Group and Symmetry wrote the following to prove a group of order 6 is isomorphic to $\mathbf{Z}_6$:
The right cosets $\langle x\rangle$, $\langle x\rangle y$ give 6 elements $e$, $x$, $x^2$, $y$, $xy$, $x^2y$ which fill out $G$.
Where $x$ is of order $3$ and $y$ is of order 2. $x$, $y$ are both elements of $G$.
My confusion:
How can one guarantee that the right coset $\langle x\rangle$ and $\langle x\rangle y$ can fill out $G$? Because $\langle x\rangle$ and $\langle x\rangle y$ have no intersections? I am not sure about the properties of cosets. I also don't know if this is related to the fact that $|\langle x\rangle|=5$ and is precisely half the elements of $G$ and thus its left coset $y\langle x\rangle$ and right coset $\langle x\rangle y$ is exactly the same?
I don't know if you can understand my question. The question here actually arises from my vague understanding of "cosets". So everytime this term occurs I feel a steady uncertainty.