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Math Help - 2-sylow subgroups of S6

  1. #1
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    2-sylow subgroups of S6

    Can someone suggest some method to find 2-sylow subgroups of S6. From what I know - we first find 2-sylow subgroups of S8 and then find an isomorphism from S6 into S8. Then our desired subgroup is (S6 /\ xQx') where Q is 2-sylow subgroup of S8. This method requires us to find x, which may not be easy. IS there an alternative method?
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  2. #2
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    Solution

    The link has good detailed description on 2-sylow subgroups:
    http://sps.nus.edu.sg/~kokyiksi/roddyurops.pdf

    However, for S6 we can directly get the 2-sylow subgroup.
    Consider, S4. The 2-sylow subgroup in this group has order 2^3 = 8 and is given as:
    P = {(1)(2)(3)(4), (1 2), (3 4), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 4 2 3), (1 3 2 4)}.
    Since S6 has 6 characters {1, 2, 3, 4, 5, 6}. Consider:
    Q = AP. Where A = ((5 6)).
    Clearly, Q is of order 2^4 = 16 and is the required 2-sylow subgroup.
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