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

For example, the subgroups of [itex]G=Z_2\oplus Z[/itex] are easy...you 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Find subgroups of finitely generated abelian groups

**Physics Forums | Science Articles, Homework Help, Discussion**