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Alternating Groups

  1. Mar 12, 2009 #1
    Let An (the alternating group on n elements) consist of the set of all even permutations in Sn. Prove that An is indeed a subgroup of Sn and that it has index two in Sn and has order n!/2.

    First of all, I need clarification on the definition of an alternating group. My book wasn't really good in explaining it.

    My attempt at a solution is limited but I was thinking I can prove that An is a subgroup by showing that its elements are closed under composition and it is closed under inverses, but I need to better understand what an alternating group is before I start a proof. Also the order n!/2, I believe has to do with calculating permutations nPn-2 = n!/2! = n!/2.

    Thanks in advance.
     
  2. jcsd
  3. Mar 12, 2009 #2

    Mark44

    Staff: Mentor

  4. Mar 12, 2009 #3
    A_n is simply the set of all even permutations. You know, i assume, what a permutation is right?

    So, now you are asked to prove that actually the set of all even permutations, namely A_n, is a subgroup, when the opertaion is composition(multiplication). So all you need to show is that for any two elements of An, call them a,b, then ab is in A_n, that the inverse of every even permutaion is an even permutation, and you are done, siince there is a theorem that guarantees that any set that satisfies these conditions is a subgroup.
     
  5. Mar 12, 2009 #4
    thank you so much, it makes sense now
     
  6. Mar 12, 2009 #5
    Now the part of proving that the order of An is n!/2 requires more work.

    First you probbably know that the order of Sn is n!. however we cannot rightaway conclude that An has n!/2 elements. We first need to show that there are as many even permutations as ther are odd permutations in Sn and then it is clear that the ord of An is n!/2.

    Hint:

    Let

    [tex] S_n=(\alpha_1,\alpha_2,...,\alpha_k,\beta_1,\beta_2,...,\beta_r)[/tex]

    Be the set of all permutations where [tex]\alpha_i,\beta_j[/tex] are even and odd permutations respectively.

    Now, what you can do is show that [tex] \beta_1\alpha_i[/tex] for i=1,..,k are all odd permutations....it requires a proof by contradiction somewhere along the lines.

    Then again, you might want to show that [tex] \beta_1\beta_j[/tex] for j=1,...,r are all even permutations.

    What do these two things tell you?
     
    Last edited: Mar 12, 2009
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