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Math Help - A subgroup of symmetry groups

  1. #1
    Junior Member universalsandbox's Avatar
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    A subgroup of symmetry groups

    What is the largest subgroup of S_{5} that is also a primary group. (prime group) and how do you find it. What about for S_{6}.
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    Quote Originally Posted by universalsandbox View Post
    What is the largest subgroup of S_{5} that is also a primary group. (prime group) and how do you find it. What about for S_{6}.
    I am not sure what you mean by "primary". If you mean a subgroup of prime order then that is easy to answer. Such as subgroup must be generated by an element of order 5 i.e. a 5-cycle. Thus, (12345) is an element of order 5 and so \left< (12345)\right> is a primary subgroup.
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    Junior Member universalsandbox's Avatar
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    yes, I was looking for the Sylow p-subgroup of S_{5}
    that is, a subgroup of S_{5} being as large as it can be and a p-group.
    Last edited by universalsandbox; December 9th 2008 at 01:59 PM. Reason: info
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    Quote Originally Posted by universalsandbox View Post
    yes, I was looking for the Sylow p-subgroup of S_{5}
    .
    Since |S_5|=5! = 2^3\cdot 3\cdot 5 so we can look for Sylow 2,3,5 subgroups.
    For the Sylow subgroups of S_5 it has order 5 and so is cyclic. Therefore, the Sylow 5-subgroups are generated by a 5-cycle. The same with p=3. With p=2 its gets a lot more involved, you can get many dihedral subgroups for example.
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