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Math Help - Quotient group of S3

  1. #1
    Forum Admin topsquark's Avatar
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    Quotient group of S3

    It's a good thing I'm doing this review. I'm running into all kinds of stuff that I had skipped over.

    I am constructing a quotient group of S_3:
    Let
    e = \left ( \begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix} \right )

    a = \left ( \begin{matrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{matrix} \right )

    b = \left ( \begin{matrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{matrix} \right )

    c = \left ( \begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix} \right )

    d = \left ( \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix} \right )

    f = \left ( \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{matrix} \right )

    I constructed the multiplication table as usual. Now I take the subgroup E = \{ e, a \} and construct S_3 / E. I get the cosets
    E = \{ e, a \}
    A = \{ b, c \}
    B = \{ d, f \}

    Now I want to construct the multiplication table for S_3 / E, but when I apply the definition (Ex)^{-1} = Ex^{-1} I get an apparent problem:
    (Eb)^{-1} = Eb^{-1} = Ed = \{ d, f \} = B
    but
    (Ec)^{-1} = Ec^{-1} = Ec = \{ b, c \} = A

    I would have expected both answers to be B, the inverse of A under the group S_3 / E. What did I do wrong?

    Thanks!
    -Dan
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  2. #2
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    Not a Normal subgroup

    I am reviewing some Algebra as well. I THINK that this is problem

    E is a subgroup of S_3 but it is not a normal subgroup

    If A is a subgroup of G. Then A is a normal subgroup if xA=Ax for all x \in G

    Note that this is a Set equality.

    For you cE \ne Ec so E isn't normal

    Then the defintion of a Quoteint Group is

    If H is a normal subgroup of G, the group G/H that consists of the cosets of H in G is called the quotient groups.

    Conclusion: I think that E must be Normal not just a subgroup to construct a quotient group.

    I will keep looking.

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    When dealing with S_3 it is extremely helpful to let \alpha = (1,2,3) and \beta = (1,2). Then  e = \alpha^3 = \beta^2 and \alpha = d, \alpha^2 = b, \beta = f, \alpha \beta = c, \alpha^2 \beta = a. And another useful property that \beta \alpha = \alpha^2 \beta.

    Thus, S_3 = \{ e , \alpha, \alpha^2, \beta , \alpha\beta ,\alpha^2 \beta\} with the properties that: \alpha^3 = \beta^2 = 1 and \beta \alpha = \alpha^2 \beta.

    With that above you can easily multiply elements. For example, (\alpha \beta)(\alpha^2 \beta) = \alpha (\beta \alpha^2) \beta = \alpha (\beta \alpha) \alpha \beta.
    Which becomes \alpha (\alpha^2 \beta) \alpha \beta = \beta \alpha \beta = \alpha^2 \beta^2 = \alpha^2.

    Also the subgroup diagram is given below.

    You wish to form S_3/E where E= \{e ,a\} = \{ e, \alpha^2 \beta \} but in order to do that you need to know if E is a normal subgroup. Let us see if the left cosets agree with the right coset if we use \alpha. Note \alpha E = \{ \alpha , \beta\} while E\alpha = \{ \alpha, \alpha \beta \}. Which is a problem.
    Attached Thumbnails Attached Thumbnails Quotient group of S3-conformal.jpg  
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  4. #4
    Forum Admin topsquark's Avatar
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    Once again I missed that tiny little word "normal." As a lame defense I will mention that my Physics classes apparently used normal subgroups all the time, but never mention them as being that.

    -Dan
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    Quote Originally Posted by topsquark View Post
    Once again I missed that tiny little word "normal." As a lame defense I will mention that my Physics classes apparently used normal subgroups all the time, but never mention them as being that.
    Maybe that is because in Physics classes you use abelian groups all the time?
    (Which automatically makes them normal).
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Maybe that is because in Physics classes you use abelian groups all the time?
    (Which automatically makes them normal).
    Actually most of my work where it might have mattered is with simple Lie groups. (Perhaps I should say "the simpler Lie groups." For all I know there is a specific definition for "simple" in reference to Lie groups.) Mostly it has been with the differential structure of the group rather than global, so this is probably why it hasn't been an issue. But for the record much of my work has been with U(1) and SU(2) (with only a little SO(3) and SU(3) thrown in).

    Thanks again for the response.

    -Dan
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