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Group generated by Z_p and Z_q ?

  1. Nov 13, 2007 #1
    [tex]\mathbb{Z}_{p}[/tex] and [tex]\mathbb{Z}_{q}[/tex] are (within themselves) abelian by the fact they each only have one generator, say [tex]\omega_{p}[/tex] and [tex]\omega_{q}[/tex]. However, when combined, they aren't (in general).

    Is there a systematic way of generating all the different elements in the group with generators [tex]\omega_{p}[/tex] and [tex]\omega_{q}[/tex]? I'm currently only working with [tex]\mathbb{Z}_{2}[/tex] and [tex]\mathbb{Z}_{3}[/tex], but there's 12 elements in the group generated by both of them and if I do it in a combinatorical way I get huge amounts of the same elements in my end list.

    Is there a way to stream line it a bit so that minimal repetition of the same elements occurs? Obviously I can do it by hand for my example (and I did) but I'm looking to automate it for groups up to [tex]\mathbb{Z}_{12}[/tex] and that'll be out of the question by hand and unless there's a nice way, computationally intensive.

    Thanks for any help :)
     
  2. jcsd
  3. Nov 13, 2007 #2

    morphism

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    How are you "combining" these two groups?
     
  4. Nov 13, 2007 #3
    I'm considering the action of these groups on a 6d torus so it comes down to saying a symmetry alters the 6 coordinates. Specifically :

    [tex]\mathbb{Z}_{2} \quad \theta &:& (x^{1},y^{1},x^{2},y^{2},x^{3},y^{3}) \to (x^{1},y^{1},-x^{2},-y^{2},-x^{3},-y^{3}) [/tex]
    [tex]\mathbb{Z}_{3} \quad \phi &:& (x^{1},y^{1},x^{2},y^{2},x^{3},y^{3}) \to (x^{3},y^{3},x^{1},y^{1},x^{2},y^{2}) [/tex]

    Therefore I can write them as matrices (with each '1' being the 2x2 identity matrix) :

    [tex]\theta = \left( \begin{array}{ccc}
    1 \\ & -1 \\ & & -1
    \end{array} \right) [/tex]
    [tex]\phi = \left( \begin{array}{ccc}
    \quad & \quad & 1 \\ 1 \\ & 1 & \quad
    \end{array} \right)[/tex]

    The various ways of combining [tex]\theta[/tex] and [tex]\phi[/tex] then lead to 12 different matrixes. [tex]\phi[/tex] gives 3 different layouts for the non-zero entries and depending on where you put [tex]\theta[/tex] in the matrix operator sequence, you shuffle around where the 1's and -1's are (there's 4 different ones for each layout).

    At present I'm just computing all combinations of [tex]\theta[/tex] and [tex]\phi[/tex] which go up to length 5 (since I'm working with Z_2 and Z_3) and that covers everything.
     
    Last edited: Nov 13, 2007
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