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Math Help - symmetry of plane figures question

  1. #1
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    symmetry of plane figures question

    Let G be a finite group of rotations of the plane around the origin, prove that G is cyclic.

    Im confused about how to start this (I have not done any group theory in a while)

    Thanks for any help

    Take the smallest rotation, denoted as g, then there exists a least m such that g^m=1

    As G is a finite group |G|=n

    If m=n then we are done. \{1,g,g^2,.....,g^{m-1}\}

    If m<n

    then there exists an element b\in G s.t. b\neq g^i

    do I then go on to show that if this was the case b must have an inverse and that would be one of the g^i's the we have a contradiction?
    Last edited by hmmmm; July 11th 2011 at 08:44 AM.
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  2. #2
    Senior Member abhishekkgp's Avatar
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    Re: symmetry of plane figures question

    Quote Originally Posted by hmmmm View Post
    Let G be a finite group of rotations of the plane around the origin, prove that G is cyclic.

    Im confused about how to start this (I have not done any group theory in a while)

    Thanks for any help

    Take the smallest rotation, denoted as g, then there exists a least m such that g^m=1

    If m=n then we are done. \{1,g,g^2,.....,g^{m-1}\}
    what is n??
    If m<n

    then there exists an element b\in G s.t. b\neq g^i

    do I then go on to show that if this was the case b must have an inverse and that would be one of the g^i's the we have a contradiction?
    ...
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  3. #3
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    Re: symmetry of plane figures question

    oops sorry it is the order of the group G, but i think what I have done may be wrong....
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    Senior Member abhishekkgp's Avatar
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    Re: symmetry of plane figures question

    Quote Originally Posted by hmmmm View Post
    oops sorry it is the order of the group G, but i think what I have done may be wrong....
    well if that's what n is then m has to be equal to n because you said that g represents the smallest rotation.
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  5. #5
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    Re: symmetry of plane figures question

    Im sorry im a bit confused by what you have said, if n=m then I am done, could you possibly give more explanation?

    thanks for any help
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Re: symmetry of plane figures question

    Quote Originally Posted by hmmmm View Post
    Let G be a finite group of rotations of the plane around the origin, prove that G is cyclic.

    Im confused about how to start this (I have not done any group theory in a while)

    Thanks for any help

    Take the smallest rotation, denoted as g, then there exists a least m such that g^m=1

    As G is a finite group |G|=n

    If m=n then we are done. \{1,g,g^2,.....,g^{m-1}\}

    If m<n

    then there exists an element b\in G s.t. b\neq g^i

    do I then go on to show that if this was the case b must have an inverse and that would be one of the g^i's the we have a contradiction?
    I assume you know that every rotation of the plane is of the form \begin{pmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{pmatrix}. It's easy then to see that the map \begin{pmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{pmatrix}\mapsto e^{i\theta} defines an embedding (injective homomorphism) G\hookrightarrow \mathbb{C}^\times. Thus, it suffices to prove that every finite subgroup H of \mathbb{C}^\times is cyclic. To do this we merely note that by Lagrange's theorem x^{|H|}-1=0 for all x\in H and by the FTA we know that x^{|H|}-1 has precisely |H| roots, and moreover we know they are the |H| roots of unity which are trivially cyclic.
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  7. #7
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    Re: symmetry of plane figures question

    Thanks, not the way that I think I was meant to do this but I think I get it (this seems a little more complicated then what I was expecting).

    Thanks very much.
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Re: symmetry of plane figures question

    Quote Originally Posted by hmmmm View Post
    Thanks, not the way that I think I was meant to do this but I think I get it (this seems a little more complicated then what I was expecting).

    Thanks very much.
    You can do the minimum angle whosuwutsit if you want or you can try doing a similar proof to mine except using characteristic or minimal polynomials.
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  9. #9
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    Re: symmetry of plane figures question

    no your proof is much nicer than what I was attempting to do. all I was saying was that I had not thought of doing that at all. thanks very much!
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