- #1
Prof. 27
- 50
- 1
Homework Statement
Find all cosets of the subgroup H in the group G given below. What is the index (G : H)?
H = <(3,2,1)>, G = S3
Homework Equations
The Attempt at a Solution
I will leave out the initial (1,2,3) part of the permutation. We have S3 = {(1,2,3),(2,1,3),(3,2,1),(3,1,2),(2,3,1),(1,3,2)}
And for H we have
(3,2,1)
(3,2,1)+(3,2,1) = (3,1,2)
(3,2,1)+(3,2,1)+(3,2,1) = (3,3,3)
So H = {(3,2,1),(3,1,2),(3,3,3)}
The problem is that (3,3,3) is not in S3. If I ignore this then I find the cosets:
0 + <(3,2,1)> = {(3,2,1),(3,1,2),(3,3,3)}
1 + <(3,2,1)> = {(1,3,2),(1,2,3),(1,1,1)}
2+ <(3,2,1)> = {(2,1,3),(2,3,1),(2,2,2)}
This exhausts S3 but there are these additional elements not in it. I can't figure out what I'm missing. Any pointers?