EDIT:( Sorry I meant for the title of this to say conjugacy classes of subgroups of Z X Z) I have a question.(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to figure out, what are all of the normal subgroups of Z X Z? Well, I know that Z has the following subgroups: the trivial group {0}, itself Z, and then aZ, one for each a in Z. (the group of all multiples of an integer a).

Well, this makes me assume that Z X Z has the following subgroups:

0, Z X Z, Z X aZ (one for each positive integer a), bZ X Z (one for each positive integer b) and cZ X dZ (one for each set of positive integers c,d)

First off, I was wondering is this logic correct? Is there a relevant theorem which says that if you have a cross of two groups, that the subgroups of the new cross product is the cross of its subgroups?

Also, am I right in assuming that all of the groups of the form Z X aZ, cZ X dZ and bZ X Z are in the same conjugacy class?

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# Normal subgroups of Z X Z

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