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?