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Finding Subgroups of certain orders

  1. Nov 19, 2011 #1
    I'm not sure how for example I would go about finding the subgroups of S_4 of order 4?

    I don't quite understand how I would I find or construct subgroups for a permutation group?

    Any advice?

  2. jcsd
  3. Nov 19, 2011 #2


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    1 ->2 ->3 ->4 ->1
    1 ->3 ->2 ->4 ->1 for example
  4. Nov 19, 2011 #3


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    Suppose [itex]\sigma \in S_n[/itex] is a 4-cycle. What can you say about [itex]\langle \sigma \rangle[/itex]?
  5. Nov 19, 2011 #4
    After using both your advice I think I have made some progress.

    I used (1 2 3 4) and generated a subgroup of order 4, this subgroup was, {id, (1234), (13)(24), (1432)}, is this correct? And this is of order 4 so meets the condition.
    I also did the same with (1324).

    After reading a theorem I concluded if I want the subgroups to be of order 4 then the elements of the subgroups must divide 4. So since I've done the two for 4 would I know need to do the same for elements of order 2? Is there a shortcut or do I have to compute them all to check they form subgroups of order 4?

    After attempting this with for example (12)(34) I found the usual method of multiplying by itself to try and find a group quickly failed since (12)(34)(12)(34) is simply id.

    How do I find the subgroups using elements of order 2?

    Thanks for the previous advice and an advance thanks for anymore
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