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) :

$$\theta = \left( \begin{array}{ccc} 1 \\ & -1 \\ & & -1 \end{array} \right)$$
$$\phi = \left( \begin{array}{ccc} \quad & \quad & 1 \\ 1 \\ & 1 & \quad \end{array} \right)$$

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.