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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


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

  4. Jun 27, 2011 #3

    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


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

  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


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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
    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


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


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