Question

List all left cosets of the subgroup $LaTeX: 3\mathbb{Z}$ of $LaTeX: \mathbb{Z}$ and find the index .

Answer 1

z = {... - 3, - 2, - 1, 0, 1, 2,3,...} 3z = {... - 9, - 6, - 3, 0, 3, 6, 9,...} 0 + 3z = {... - 9, - 6, - 3, 0, 3,6,9..} 1 + 3z = {... - 8, - 5, - 2, 1, 4, 7, 10,...} 2 + 3z = {... - 7, - 4, - 1, 2, 5, 8, 11,...} If m be any integer other than 0,1,2 there m + 3z = {3z if m = 3k, k elementof z. 1 + 3z if m = 3k + 1, k elementof z. 2 + 3z if m = 3k + 2, k elementof z Hence 3z, 1 + 3z, 2 + 3z are distinct left of 3z in z. (z: 3z) = Total number of distinct left of 3z in z. Therefore (z: 3z) = 3.