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Let $H = \{\text{id}, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) \} \subset S_4$, and let $ K = \{\sigma \in S_4 \mid \sigma(4) = 4\}.$

(a) How to show that $H$ is a subgroup of $S_4$, and is $H$ is a normal subgroup? What other group is $H$ isomorphic to? Same for $K.$

(b) Does every coset of $H$ contain exactly one element of $K.$ Also, can every element of $S_4$ be written uniquely as the product of an element of $H$ and an element of $K$?

(c) What can be said about the quotient group $S_4/H$? Is $S_4$ isomorphic to the direct product $H × K$?

For part a, to show that H is a subgroup of S4, I will take each element in S4 and conjugate it by what? i.e., to be normal NH = HN, or g-1ng = n

for part b, I am not sure what is being asked of

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    James, it is not considered polite here to just paste in a question and wait for answers - it does not show that you have thought about the problem. Please explain what you've tried so far, and where you are stuck.2012-03-06
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    Try drawing a picture of a square and labeling the vertices cyclically. How does H act on your picture?2012-03-06
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    What methods do you know for showing something is a subgroup? for showing something is a normal subgroup? Do you know all the groups with the same order as $H$, and how to tell them apart? Can you list the elements of $K$? Can you list the cosets of $H$?2012-03-06
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    @Zev, I did explain some stuff but I need help in understanding the question2012-03-06
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    I think Gerry Myerson shouted the answer too loud that I am hearing it right away. I would add an answer if nobody else does after a day. I hope that would be long enough or may be not.2012-03-06

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