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Math Help - permutations, cosets, and direct products

  1. #1
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    permutations, cosets, and direct products

    Can anyone help me with this problem please... thank you


    Show that for every subgroup H of Sn for n ≥ 2, either all the permutations in H are even or exactly half of them are even.
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  2. #2
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    I guess this approach might work:

    Let X be the set of all even permutations of H. Now if X = H the proof is complete. Otherwise there exists a non-empty set of odd permutations Y.

    Now you define a one-to-one onto map \phi : X \mapsto Y.
    Define \phi (\sigma) = (1,2)\sigma for \sigma \in X.

    This shows that |X| = |Y| thus there are exactly one half even permutations and one half odd permutations.
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