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Math Help - unique subgroups

  1. #1
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    unique subgroups

    Prove that A_4 has a unique subgroup V of order 4.

    I know all 12 elements of A_4, and I know V contains { 1, (12)(34), (13)(24), (14)(23)}, but how do I prove that V is a unique subgroup?
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  2. #2
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    Quote Originally Posted by dori1123
    Prove that A_4 has a unique subgroup V of order 4.

    I know all 12 elements of A_4, and I know V contains { 1, (12)(34), (13)(24), (14)(23)}, but how do I prove that V is a unique subgroup?
    Since subgroups of order 4 are Sylow 2-subgroups it follows all subgroups of order 4 must be conjugate by Sylow's second theorem. Now V is a normal subgroup of A_4. Therefore, it must be the only one.
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