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Normal Cyclic Subgroup

  1. Feb 15, 2010 #1
    1. The problem statement, all variables and given/known data

    Is the Cyclic Subgroup { (1), (123), (132)} normal in [tex]A_{4}[/tex] (alternating group of 4)

    2. Relevant equations



    3. The attempt at a solution

    So I believe if I just check if gH=Hg for all g in A_4 that would be suffice to show that it is a normal subgroup, but that seems really tedious. Is there a easier way?

    Also how can I figure out what the elements of A_4 are? I know its the even permutations but is there a way to quickly identity which ones it is? How do I visualize it?
     
  2. jcsd
  3. Feb 15, 2010 #2
    Visually speaking, A_4 is the group associated with rotations of the regular tetrahedron, if that helps.

    One can use the Sylow theorems to prove normality sometimes, but in this case it doesn't help.
     
  4. Feb 15, 2010 #3
    So only way is by brute force?
     
  5. Feb 15, 2010 #4

    Dick

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    Homework Helper

    No, it's not the only way. In this case, I would try and guess an element g of A4 such that gHg^(-1) is NOT equal to H, where H is your subgroup. It's not hard. You can recognize whether a permutation is even just by looking at it's cycle structure. Cycles with an odd number of elements are even permutations and cycles with an even number of elements are odd permutations.
     
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