List the 4 cosets of 1132133Z 34SolutionConsider the group H

List the 4 cosets of {1,13,21,33}Z 34.

Solution

Consider the group H = {1, 13, 21, 33}

Z₃₄* means the multiplication operator modulo 34

Now perform 33H or { 33, 33*13, 33*21, 33*33} modulo 34

= { 33 mod 34, 429mod34, 693mod34, 1089mod34} = {33, 21, 13,1 } = H

EDIT: Subsets or subgroups have to satisfy group criteria.

1) Has identity element a.1modk = a mod k => 1 is identity.

2) The order or number of elements have to be a divisor of order of the group.

No of elements in H is 4. So, subgroup can have order = 2 and 4. Also, each subgroup should have identity. That\'s how we arrive at order 2 subsets of H are A = {1,13}, B = {1,21,} and C = {1,33}

A² = {1,169} mod 34 = {1, 33} = C, B² = {1, 441} mod34 = { 1,33} = C => A and B are also cosets

Cosets are {1}, A, B and H or {1}, {1,13}, {1,21} and {1,13,21,33}