- #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 = S

_{3}

## Homework Equations

## The Attempt at a Solution

I will leave out the initial (1,2,3) part of the permutation. We have S

_{3}= {(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 S

_{3}. 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 S

_{3}but there are these additional elements not in it. I can't figure out what I'm missing. Any pointers?