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Permutations - determine order of S_n

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known datahttp:
    What is the largest number which is the order of an element of S_8? Write down
    an element of that order in disjoint cycle notation.


    2. Relevant equations



    3. The attempt at a solution
    To start with, I don't understand the wording of the question. When it refers to element, does it imply permutation. If so, then is the question asking that we find the composition of S_8 such that the order is at its largest (wherbey the order is the product of the least common multiple of the cycle lengths)?

    For ex,
    (12345)(678)
    The order is 15
     
  2. jcsd
  3. Feb 7, 2010 #2
    Recall that the order of an element of a group is the order of the group generated by that element. Equivalently, if a is an element of some group, then its order is the smallest n such that [itex]a^n=e[/itex], where e is the identity.
     
  4. Feb 7, 2010 #3
    So you're suggesting that the largest order for S_8 is 8?

    But can it be 15? Referring back to my example...
     
  5. Feb 7, 2010 #4
    Elements of [tex]S_8[/tex] are permutations, as you mentioned. Each element of [tex]S_8[/tex] has an order, of all possible orders of elements, the question asks for the largest. I think you would be right that it is in fact 15.
     
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