Check that the left cosets of the subgroup K e 123 132 in S

Check that the left cosets of the subgroup K = {e, (123), (132)} in S_3 are e K = (123)K = (132)K = K (12)K = (13)K = (23)K = {(12), (13), (23)} and that each occurs three times in the list (gK)g S_3. Note that K is the subgroup of even permutations and the other coset of K is the set of odd permutations.

Solution

Given that

K = {e, (123), (132)}

eK = {e*e, e(123) e(132)} = {e, (123), (132)} =K

(123)K={(123)e, (123)(123), (123)(132)}= {(123), e, (132)} or {(123), (132) e} = K (due to closure in K)

Similarly

(132)K = K