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Math Help - Groups complex

  1. #1
    Newbie Belzelga's Avatar
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    Apr 2008
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    Groups complex

    Hello

    I got 2 questions, which seems hard to follow any ideas in how to start off?

    Set
    S of generators of group G is called a minimal set of generators if
    for all
    s 2 S, the set S n fsg is no longer a set of generators. Find a minimal

    set of generators for
    D6

    and this one is complex

    Use complex numbers to prove that cos
    4(beta)+ sin4 (beta)=(1/4)(3 + cos 4(beta)).

    Any help is appreciated
    Thank you

    Belzelga

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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Belzelga View Post
    Hello

    I got 2 questions, which seems hard to follow any ideas in how to start off?

    Set
    S of generators of group G is called a minimal set of generators if
    for all
    s 2 S, the set S n fsg is no longer a set of generators. Find a minimal

    set of generators for
    D6

    and this one is complex

    Use complex numbers to prove that cos
    4(beta)+ sin4 (beta)=(1/4)(3 + cos 4(beta)).

    Any help is appreciated
    Thank you

    Belzelga
    For the first problem, there are two competing meanings for the notation D_6. It is a dihedral group, with eith 6 elements or 12 elements, according to which definition you use. In either case, it can be described by two generators (a rotation and a reflection). This is a minimal set, because of you remove one generator then the remaining single generator could only generate an abelian group. But whichever way you define the group D_6, it is nonabelian.

    I think that the second problem should read \cos^4\beta + \sin^4\beta = {\textstyle\frac14}(3+\cos(4\beta)). The way to prove this is to put z=cos(β)+i.sin(β). Then, by de Moivre's theorem, z^4 = \cos(4\beta) + i\sin(4\beta). But z^4 = (\cos\beta+i\sin\beta)^4 = \cos^4\beta + 4i\cos^3\beta\sin\beta -6\cos^2\beta\sin^2\beta -4i\cos\beta\sin^3\beta + \sin^4\beta.

    Take the real part of those expressions to see that \cos(4\beta) = \cos^4\beta -6\cos^2\beta\sin^2\beta + \sin^4\beta. But \cos^2\beta\sin^2\beta = {\textstyle\frac14}\sin^2(2\beta) = {\textstyle\frac14}\bigl({\textstyle\frac12}(1-\cos(4\beta))\bigr). Substitute that into the previous equation and you get the result.
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