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Math Help - Group theory

  1. #1
    Newbie pc31's Avatar
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    Group theory

    I have a feeling that this question should not be hard but somehow I don't really understand it.. Please help with some detail explanation if you can... (I don't just need an answer but really want to understand it...)

    Let f(x,y) = (-x,y)
    h(x,y) = (-1/2 -sqrt3/2) (x)
    (sqrt3/2 -1/2 ) (y)
    (i.e. rotate counterclockwise and angle of 2pi/3

    Define a group G = {f^k.h^j|k = 0,1 ; j = 0,1,2}
    Find a formula that expresses (f^i.h^j)*(f^s.h^t) = f^a.h^b
    Show that G is nonabelian with order 6

    Thank you so much!!!
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  2. #2
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    1) f^2 = \text{id}
    2) h^2 = \text{id}
    3) hf = f^2h

    That should be sufficient (I hope I did not make any mistake for I did that in my head).
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  3. #3
    Newbie pc31's Avatar
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    er.. i'm sorry but i dont understand your post.. do you mind elaborate a little more?
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  4. #4
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    h(x,y) = (-1/2 -sqrt3/2) (x)
    (sqrt3/2 -1/2 ) (y)
    (i.e. rotate counterclockwise and angle of 2pi/3
    That's not what your formula says. I presume you mean h(x,y)= ((-1/2- sqrt(3))x+ (sqrt(3)/2-1/2)y, (-1/2- sqrt(3))x+ (-1/2- sqrt(3)/2)y).

    What part do you not understand?

    Do you understand that f^2= id? What is f(x,y)? What is f(f(x,y))?

    Do you understand that h^2= id? What is h(x,y)? What is h(h(x,y))?

    Do you understand that hf= h^2f? What is h(f(x,y))? what is h(h(f(x,y)))?

    The problem asked you to "Find a formula that expresses (f^i.h^j)*(f^s.h^t) = f^a.h^b". Well, you have it now don't you?

    Now, write out all the different members of G.
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  5. #5
    Newbie pc31's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    That's not what your formula says. I presume you mean h(x,y)= ((-1/2- sqrt(3))x+ (sqrt(3)/2-1/2)y, (-1/2- sqrt(3))x+ (-1/2- sqrt(3)/2)y).
    You see, that's the part i dont understand... I'm sorry i was try to type in a matrix but failed. h(x,y) is the formula for rotating (x,y) and angle of 2pi/3.
    It is a 2*2 matrix, The first line has entires (-1/2) and (-sqrt3)/2. The second line has entries (sqrt3)/2 and (-1/2). and of course that times (x,y) ....

    Any help please?
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  6. #6
    Newbie pc31's Avatar
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    never mind, i know how to solve this already... thank you tho...
    everyone else who is interested, look up for dihedral group!!!!
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  7. #7
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    Quote Originally Posted by pc31 View Post
    never mind, i know how to solve this already... thank you tho...
    everyone else who is interested, look up for dihedral group!!!!
    You can generalize your problem.
    Let A be the rotation matrix of angle 2\pi/n counterclockwise.
    Let B be the reflection matrix through y-axis.

    Then G = \{ A^iB^j | 0\leq i < n, 0\leq j < 2 \} is (isomorphic) to the dihedral group D_{n}.
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