Find all cosets of {e,(1,2)} in S3

Hello!

I am hoping if someone could please explain how to do this question as I cannot get my head around it.

Q1. Find all cosets of {e,(1,2)} in S3

> How do I know what is in S3 if im not told in the question? will it always be s3={ e,(1,2),(1,3),(2,3),(1,2,3),(1,3,2)} (this is saying that there are 6 subgroups?)

> Letting H={e,(1,2)} {this is saying that this is an order of 2?)

> 6/2 = 3 therefore saying there should be 3 cosets?

My main problem is actually solving this out.

eH = {e,(1,2)} = He

(1,2)H = { (1,2)(1,2)(1,2) } = {(1,2),e}= He <- How is this calculated?

(1,3)H = { (1,3)(1,3)(1,2) } = {(1,3),(1,2,3)}= H(1,3) <- How is this calculated?

(2,3)H = { (2,3)(2,3)(1,2) } = {(2,3),(1,3,2)}= H(2,3) <- How is this calculated?

(1,2,3)H = { (1,2,3)(1,2,3)(1,2) } = {(1,2,3),(1,3)}= H(1,2,3) <- How is this calculated?

Re: Find all cosets of {e,(1,2)} in S3

Hello,

The only way I think I know how to do it, is by drawing up a s3 Cayley table and subbing in the numbers.

When I do that, i end up with;

12H = { 12, 13, 23 } = H12

13H = { 13, 23, 12 } = H13

23H = { 23, 12, 13 }

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

132H = { 132, e, 123}

Therefore, 3 cosets. Could some please confirm if this is ok for me to do it like this?