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Subgroups of size 5 in A_6

  1. Oct 31, 2009 #1
    I need to find the number of subgroups of size 5 in A_6.


    I have started by noting that as the subgroup size is 5, a prime, the subgroups must be cyclic. I have worked out that there are 144 elements of order 5 in A_6, but this cant be equal to the number of subgroups (i found two subgroups which have the same elements in!). Someone please help!
     
  2. jcsd
  3. Oct 31, 2009 #2

    Dick

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    If two subgroups of order 5 intersect, can you describe the intersection?
     
  4. Nov 1, 2009 #3
    Ok, I have noticed that <a> = <a^2> = <a^3> = <a^4> for all a in A_6 where a is a 5-cycle. So this means that the number of elements of order 5 must be divided by 4. Hence the answer is 144/4 = 36 subgroups of size 5 in A_6.
    Is this correct?
     
  5. Nov 1, 2009 #4

    Dick

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    Yes, every subgroup of order 5 contains four of them and no element of order 5 is contained in two different subgroups.
     
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