Group generated by Z_p and Z_q ?

[AlphaNumeric2]

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

Is there a systematic way of generating all the different elements in the group with generators \omega_{p} and \omega_{q}? I'm currently only working with \mathbb{Z}_{2} and \mathbb{Z}_{3}, 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 \mathbb{Z}_{12} and that'll be out of the question by hand and unless there's a nice way, computationally intensive.

Thanks for any help :)

 

[morphism]

How are you "combining" these two groups?

 

[AlphaNumeric2]

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 :

\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})
\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})

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

\theta = \left( \begin{array}{ccc}<br /> 1 \\ &amp; -1 \\ &amp; &amp; -1<br /> \end{array} \right)
\phi = \left( \begin{array}{ccc}<br /> \quad &amp; \quad &amp; 1 \\ 1 \\ &amp; 1 &amp; \quad<br /> \end{array} \right)

The various ways of combining \theta and \phi then lead to 12 different matrixes. \phi gives 3 different layouts for the non-zero entries and depending on where you put \theta 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 \theta and \phi which go up to length 5 (since I'm working with Z_2 and Z_3) and that covers everything.