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  1. #1
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    Math Help

    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; Nov 9th 2009 at 08:44 AM.
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  2. #2
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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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    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 $\displaystyle A_n$ 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 $\displaystyle \sigma,\tau\in A_n$ then $\displaystyle \sigma,\tau$ are both the product of an even number of transpositions. So argue that $\displaystyle \sigma\tau$ is also the product of an even number of transpositions.

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

    3. Let $\displaystyle \tau\in S_n$ such that $\displaystyle \tau=\prod_{i=1}^n t_i$ where $\displaystyle t_i$ are transpositions. Using 2. what is $\displaystyle \tau^{-1}$ in terms of $\displaystyle \prod_{i=1}^n t_i$? (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 $\displaystyle \sigma,\tau\in A_n$ then $\displaystyle \sigma,\tau$ are both the product of an even number of transpositions. So argue that $\displaystyle \sigma\tau$ is also the product of an even number of transpositions.

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

    3. Let $\displaystyle \tau\in S_n$ such that $\displaystyle \tau=\prod_{i=1}^n t_i$ where $\displaystyle t_i$ are transpositions. Using 2. what is $\displaystyle \tau^{-1}$ in terms of $\displaystyle \prod_{i=1}^n t_i$? (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 $\displaystyle \sigma=(ab)$ is its own inverse. Therefore $\displaystyle \sigma_e=\sigma^2=(ab)(ab)$ so that $\displaystyle A_n$ contains the identity.

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