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Math Help - Permutation subgroup

  1. #1
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    Permutation subgroup

    Prove the set of even permutation in S_{n} forms a subgroup of S_{n}.

    How should I start this?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Prove the set of even permutation in S_{n} forms a subgroup of S_{n}.

    How should I start this?
    This is referred to an "alternating group" A_n.

    How about you start off and I help you if necessary.

    Let G be a group and H a non-trivial subset of G. What makes H a subgroup? What conditions must be met?

    Now do the same thing here. Show all of these conditions are satisfied.
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  3. #3
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    Okay, let me think more about this...

    Suppose that H \subset S_{n} with H= \{ a_{1},a_{2}, ... , a_{n} \} with a being permutations that can be expressed as even number of 2-cycles, in other words, H contains all of the even permutations in Sn.

    Now pick  \alpha , \beta \in H.

    Then  \alpha = (a_{1} a_{2})(a_{3}a_{4}) and  \beta = (b_{1}b_{2})(b_{3}b_{4})

    Now,  \alpha \beta is in H because it is only another expression of even permutations. (Do I need to justify this? Or do we already know that? I know that by the problems that I have done before)

    Then  \alpha is its own inverse, thus satisfied the conditions for H to be a subgroup of Sn.

    Am I doing something right here?

    Thanks.
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  4. #4
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    [QUOTE=tttcomrader;69480]
    Am I doing something right here?
    .
    Almost.

    \alpha is even and \beta is even then \alpha \beta is even. You mentioned that but you did not show it. Why is that? Remember you need to show this to show that H is closed.
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