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    Need help proving that even permutations form a group?

    where the say of even permutations in symmetric group of degree n(S sub n) forms a subgroup of S sub n.
    Last edited by biggybarks; November 9th 2009 at 08:44 AM.
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    Quote Originally Posted by biggybarks View Post
    Need help proving that even permutations form a group?

    where the say of even permutations in symmetric group of degree n(S sub n) forms a subgroup of S sub n.
    .
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  3. #3
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    Quote Originally Posted by biggybarks View Post
    Need help proving that even permutations form a group?

    where the say of even permutations in symmetric group of degree n(S sub n) forms a subgroup of S sub n.

    What have you done, where are you stuck? The group of even permutations in n objects is denoted A_n and called the alternating group.

    Tonio
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    math help

    having a hard time getting started on it. Don't know where to begin
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    MHF Contributor Drexel28's Avatar
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    Here is a rough outline of the proof. Fill in the blanks.

    1. Suppose that \sigma,\tau\in A_n then \sigma,\tau are both the product of an even number of transpositions. So argue that \sigma\tau is also the product of an even number of transpositions.

    2. Let \sigma\in S_n be a transposition. What is \sigma^2? What does that imply?

    3. Let \tau\in S_n such that \tau=\prod_{i=1}^n t_i where t_i are transpositions. Using 2. what is \tau^{-1} in terms of \prod_{i=1}^n t_i? (if this doesn't make sense...try doing some small example and see the pattern)
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    help

    still lost. Don't really understand how to get this done
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Here is a rough outline of the proof. Fill in the blanks.

    1. Suppose that \sigma,\tau\in A_n then \sigma,\tau are both the product of an even number of transpositions. So argue that \sigma\tau is also the product of an even number of transpositions.

    2. Let \sigma\in S_n be a transposition. What is \sigma^2? What does that imply?

    3. Let \tau\in S_n such that \tau=\prod_{i=1}^n t_i where t_i are transpositions. Using 2. what is \tau^{-1} in terms of \prod_{i=1}^n t_i? (if this doesn't make sense...try doing some small example and see the pattern)
    Quote Originally Posted by biggybarks View Post
    still lost. Don't really understand how to get this done
    I think you should be able to handle one. For the second one merely note that a transpotition \sigma=(ab) is its own inverse. Therefore \sigma_e=\sigma^2=(ab)(ab) so that A_n contains the identity.

    If \tau=t_1\cdots t_n are are transpositions then \tau^{-1}=t_n\cdots t_1. To see this merely note that t_1\cdots t_n\cdot t_n\cdots t_1=t_1\cdots t_{n-1}\cdot t_{n-1}\cdots t_1=\cdots=t_1\cdot t_1=e. Therefore an even permutation has...........what?
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