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I am in an Abstract Algebra course in graduate school and am trying to prove that the symmetric group S_4 is isomorphic to a subgroup of the alternating group A_6.
I am thinking that I need to find an isomorphism between the two. It is easy to find a map that takes all the even elements of S_4 into A_6, but I am having trouble mapping the odd elements.
Thank you!
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Does anyone have any ideas on this? It would be really helpful.
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Consider the two groups as permutations of points on a plane or something.
You could visualise A6 as all the even permutations of the points of a hexagon.
Take 4 of the points on a hexagon, then can you get all permutations of those 4 points in A6?
All the even permutations are obvious, and the odd ones can be made even by swapping the extra 2 points you didn't choose!
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