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I have an exercise where I am supposed to find the left and right cosets of H = {(1), (12), (34), (12) ○ (34)} in S4. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set H and multiply it with every number in H. But this does not exactly give the right answer. I would really appreciate it if someone gave an easy to understand explanation of how to generate the left and right cosets.

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What is Ž above in your subgroup? –  B. S. May 26 '12 at 15:20
    
You don't pick a "number", you pick an element $g$ of your group $G$ ($G = S_4$ in your example). The left coset $gH$ consists of the elements $gh$ as $h$ ranges over all elements of $H$. If you are having trouble with this, you should show us what specific calculations you have done and we can help pinpoint where you might have went astray. –  Michael Joyce May 26 '12 at 15:35
    
The Ž was supposed to be an . The calculations I have done so far is selecting (13) as an element and multiplying it with H. Then I get (13) first(obviously), but if I multiply (13) with (12) things doesn't exacyly become the answer I am looking for. So I would like to know how you should do all the calculations correct, after selecting an element. –  user1049697 May 26 '12 at 15:42
    
What permutation do you get when you compute $(13)(12)$? The set $(13) H = \{ (13), (13)(12), (13)(34), (13)(12)(34) \}$ is one of the six left cosets of $H$ in $S_4$, regardless of what predetermined answer you are looking for. My guess is that you are either multiplying permutations incorrectly or your answer has misidentified left and right cosets. –  Michael Joyce May 26 '12 at 15:49
    
I think I have some problems understanding how the multiplications are done... If you take 13 and multiply it with 12 and get 156, then that does not exactly seem right. How do you do the multiplications correctly? And thanks for all the help everyone! It's really appreciated! –  user1049697 May 27 '12 at 9:44

2 Answers 2

up vote 3 down vote accepted

Assume your group is $G$ and the subgroup is $H$. By definition $gH$={$gh$|$h\in H$} is a left coset of $H$ respect to $g$, in $G$ and $Hg$={$hg$|$h\in H$} is a right coset of $H$ respect to $g$, in $G$. Here your group is $S_4$={$(),(3,4),(2,3),(2,3,4),(2,4,3),(2,4), (1,2),(1,2)(3,4),(1,2,3),(1,2,3,4),(1,2,4,3),(1,2,4),(1,3,2),(1,3,4,2),(1,3), (1,3,4),(1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3)$ }, and the $H$ is as you pointed. According to Group theory, the number of right cosets of a subgroup in its group called index is $\frac{|G|}{|H|}$. $|S_4|=4!$ and $|H|=|\langle(1,2),(3,4)\rangle|=4$ so you have atlast $\frac{4!}{4}=6$ cosets right or left for the subgroup. Here there is no matter what $g$ is taken in group $G$. For example, if you take $(1,2,4,3)$ in group, then $(1,2,4,3)H$={$(1,2,4,3)(),(1,2,4,3)(1,2),(1,2,4,3)(3,4),(1,2,4,3)(1,2)(3,4)$}. Hope to help.

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Very thorough, listing all the elements! +1 –  amWhy Mar 5 '13 at 0:19

Take for example $\,\pi:= (123)\,$ :$$\pi(1)=\pi\,,\,\pi(12)=(13)\,,\,\pi(34)=(1234)\,,\,\pi(12)(34)=(134)\Longrightarrow \pi H=\{(123),\,(13),\,(1234),\,(134)\}$$

Now try to find $\,H\pi\,$, and check whether you can find examples of $\,\pi\sigma^{-1}\in H\,$ , since then

$\,\pi H=\sigma H\,$ and you can save quite some time.

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