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How many elements of order 2 are contained in S_4?

  1. Jun 27, 2011 #1
    1. The problem statement, all variables and given/known data
    How many elements of order 2 does the symmetric group [itex]S_4[/itex] contain?


    2. Relevant equations



    3. The attempt at a solution
    I know that transpositions have order two. Moreover, any k-cycle has order k. Thus there are six elements with order two contained in [itex]S_4[/itex]?

    Is this all? Seems too simple.
     
  2. jcsd
  3. Jun 27, 2011 #2

    ehild

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    How many elements are there in S4 and what are they?

    ehild
     
  4. Jun 27, 2011 #3
    ehild,

    There are 4! = 24 elements in [itex]S_4[/itex]. The elements are all such permutations of {1,2,3,4}.

    Here is how I see the problem:

    Let [itex]\sigma \in S_4[/itex]. Then either:
    1. [itex]\sigma = (a_1 a_2)[/itex] and [itex](a_1 a_2)^2 = e[/itex] [itex]\Rightarrow[/itex] order 2.
    2. [itex]\sigma = (a_1 a_2 a_3[/itex] and [itex](a_1 a_2 a_3)^3 =e[/itex] [itex]\Rightarrow[/itex] order 3.
    3. [itex]\sigma = (a_1 a_2 a_3 a_4)[/itex] and [itex](a_1 a_2 a_3 a_4)^4 = e[/itex] [itex]\Rightarrow[/itex] order 4.

    Thus there are 6 permutations of order 2 in [itex]S_4[/itex]. Is this not correct?
     
    Last edited: Jun 27, 2011
  5. Jun 27, 2011 #4

    ehild

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    Oh, you mean the permutation group? I thought it was the point group called S4 - sorry.

    ehild
     
  6. Jun 27, 2011 #5

    I like Serena

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    No, this is not correct.
    Did you find how many of order 3 and 4 there are?
    Do they add up to 24?
     
  7. Jun 27, 2011 #6

    ehild

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  8. Jun 27, 2011 #7
    well, I got an idea but am not sure if I'm right cuz I haven't had abstract algebra yet. well, what you're claiming is that in a cyclic group G an element in G is a transposition if and only if It is of order 2. well, It's obvious that any transposition element in G is of order 2 but can we say that any element of order 2 is a transposition?
    if yes, then your question would become that in how many ways we can permute two letters from n letters keeping the others the same position they are. That would be an easy problem in combinatorics and discrete math.
     
  9. Jun 27, 2011 #8

    micromass

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    Hi Samuelb88! :smile:

    You are certainly correct that there are 6 transpositions in S4, i.e. there are 6 elements of the form (a b). These are

    [tex](1~2),(1~3),(1~4),(2~3),(2~4),(3~4)[/tex]

    However, these are not the only elements of order 2!! For example

    [tex](1~2)(3~4)[/tex]

    is also of order 2, so you got to count this one too!!
     
  10. Jun 27, 2011 #9
    Oh right! So that means there are six transpositions and three disjoint transpositions in [itex]S_4[\itex].
     
  11. Jun 27, 2011 #10

    micromass

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    Yes, that sounds right!
     
  12. Jun 27, 2011 #11
    Thanks guys!
     
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