## Coset Representatives

Let $G=\mathbb{Z}_4 \oplus U(5)$, $H=\{ (0,1),(2,1),(0,4),(2,4) \}$ and $K=\left\langle (1,4) \right\rangle = \{ (0,1),(1,4),(2,1),(3,4) \}$.

Could someone please show me how to find the cosets of $H$ in $G$, and the cosets of $K$ in $G$?

This is the answer given in the book, but I don't know how to get this answer:

$G/H = \{ H,(1,1)+H, (2,2)+H, (3,3)+H \}$

$G/K = \{ K,(0,2)+K, (0,3)+K, (0,4)+K \}$

Of course, I know by Lagrange's theorem that there must be $|G|/|H|=|G|/|K| = 16/4=4$ cosets in both groups. In $G/H$ for each coset we use the notation $a+H$, where $a$ is called the coset representative of $a+H$. So how do we go about choosing these coset representatives?

Can't we just choose any element from from $G$? Why did they choose those specific elements for those cosets of each group?