Find subgroups of finitely generated abelian groups

  1. Is there an "easy" method to finding subgroups of finitely generated abelian groups using the First Isomorphism Theorem? I seem to remember something like this but I can't quite get it.

    For example, the subgroups of [itex]G=Z_2\oplus Z[/itex] are only have [itex]0\oplus nZ[/itex] and [itex]Z_2\oplus nZ[/itex] for [itex]n\geq 0.[/itex]

    But if you have a different group, say [itex]G=Z_6\oplus Z_4[/itex], it's possible the subgroups aren't of the form [itex]<a>\oplus<b>[/itex] correct? Like <(2,2)>.

    How would you describe all the subgroups? I can do it by brute force..I'm looking for an quick easier asnwer if one exists...even in only some situations

    EDIT: maybe this makes more sense if I only need to know subgroups of a specific index?
    Last edited: Jul 19, 2011
  2. jcsd
  3. micromass

    micromass 20,076
    Staff Emeritus
    Science Advisor
    Education Advisor

    It seems you're looking for the subgroup lattice of finitely generated abelian groups?

    Well, the following article may help:

    Also keep in mind that if G and H are groups such that gcd(|G|,|H|)=1, then
    [itex]Sub(G\times H)\cong Sub(G)\times Sub(H)[/itex]

    So in your example
    [tex]Sub(\mathbb{Z}_6\times \mathbb{Z}_4)\cong Sub(\mathbb{Z}_3)\times Sub(\mathbb{Z}_2\times \mathbb{Z}_4)[/tex]

    so you only need to find the subgroups of [itex]\mathbb{Z}_2\times \mathbb{Z}_4[/itex]. The cyclic subgroups of this group are
    so all the subgroups are just products of the above groups.
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