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

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

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

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

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

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

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