[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).(adsbygoogle = window.adsbygoogle || []).push({});

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 :)

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# Group generated by Z_p and Z_q ?

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