Lagrange's theorem

Suppose G = {0,1,2,3,4,5,6,7,8,9,10,11} = Z/12Z and H = {0,4,8}.

H is a subset of G.

0○H = {0,4,8} = H

1○H = {1,5,9}

2○H = {2,6,10}

3○H = {3,7,11}

Each of them is a coset of H in G.

(these cosets are denoted by G/H, G/H = {0○H, 1○H, 2○H, 3○H})

We can write G as 0○H + 1○H + 2○H + 3○H.

Note that the coefficients 0, 1, 2, and 3 represent mod 4.

|G|/|H| = 12/3 = 4 = the index of H in G. written as [G:H]

Thus　|G| = [G:H]|H|.