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Class Equation

  1. Feb 22, 2008 #1
    hi, any hints to give me a good idea on how to solve this question would be greatly appreciated:

    Question:
    let G be a group acting on itself by conjugation, [itex] g . x = gxg^{-1}[/itex]

    Describe the set-theorem and numerical forms of the Class Equation for these actions explicitly when

    (i) [itex] G = A_4 [/itex]
    (ii) [itex] G = D_8 [/itex]
    (iii) [itex] G = D_{10} [/itex]

    thanx :)
     
  2. jcsd
  3. Feb 23, 2008 #2
    ok let me try part (ii) [itex]G = D_8[/itex]

    my working:
    ------------------------------------------------------------------------
    Firstly [itex]D_8 = [1,x,x^2,x^3,y,xy,x^2y,x^3y][/itex]
    [itex]x^4 = 1, y^2 = 1, yx = x^3y[/itex]

    If we calculate the orbits then we have:
    orbit of <1> = {1}

    <x>:
    [itex](1)x(1^{-1}) = x[/itex],
    [itex](x)x(x)^{-1} = x[/itex],
    [itex](x^2)x(x^2)^{-1} = x[/itex],
    [itex](x^3)x(x^3)^{-1} = x[/itex],
    [itex](y)x(y)^{-1} = x^2yy=x^2[/itex],
    [itex](xy)x(xy)^{-1} = xyxx^3y=xx^2yx^3y=x^6=x^2[/itex],
    [itex](x^2y)x(x^2y)^{-1} = x^2yxx^2y=x^2x^2yx^2y=x^4yx^2y=x^6yy=x^6y^2=x^6=x^2[/itex],
    [itex](x^3y)x(x^3y)^{-1} = x^3yxxy=x^3x^2yxy=x^3x^2x^2yy=x^7y^2=x^7=x^3[/itex],
    so orbit of <x> = <x^2> = <x^3> = {[itex]x,x^2,x^3[/itex]}

    similarly,
    <y>:
    [itex](1)y(1^{-1}) = y[/itex],
    [itex](x)y(x)^{-1} = x^3y[/itex],
    [itex](x^2)y(x^2)^{-1} = x^2y[/itex],
    [itex](x^3)y(x^3)^{-1} = xy[/itex],
    [itex](y)y(y)^{-1} = y[/itex],
    [itex](xy)y(xy)^{-1} = x^3y[/itex],
    [itex](x^2y)y(x^2y)^{-1} = x^2y[/itex],
    [itex](x^3y)y(x^3y)^{-1} =xy[/itex],
    so orbit of <y> = <xy> <x^2y> = <x^3y> = {[itex]y,xy,x^2y,x^3y[/itex]}

    concluding, the if [itex]D_8[/itex] cuts on itself by conjugation, the orbits are:
    <1> = {1}
    <x> = <[itex]x^2[/itex]> = <[itex]x^3[/itex]> = {[itex]x,x^2,x^3[/itex]}
    <y> = <[itex]xy[/itex]> = <[itex]x^2y[/itex]> = <[itex]x^3y[/itex]> = {[itex]y,xy,x^2y,x^3y[/itex]}
    ------------------------------------------------------------------------

    i believe i've done the hard part: so from here how would i descibe explicitly, the set theorem and numerical forms of the class equation for D_8 ? thanks :)
     
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