# Applications of the classification of finitely generated abelian groups

The Theorem of Finitely Generated Groups is well known and is a good tool that allows one to find the possible structures of a abelian group.

I would like to know if there are some other interesting applications of this theorem.

I known that this result can be generalized to modules over a principal ideal domains, but I am looking for applications of the statement for groups.

The only one I known is the converse for the the Lagrange theorem, that is very standard.

• I've never heard of "Theorem of Finitely Generated Groups." Please state it. Are you referring to the classification of finitely generated abelian groups? Feb 18 at 4:50
• Yes, the classification theorem..For a abelian group $A$, there exist $s$ and $d_1\mid\dots \mid d_r$ such that $A\cong \mathbb{Z}^s\oplus \mathbb{Z}_{d_1}\oplus\cdots\oplus\mathbb{Z}_{d_r}$. Feb 18 at 13:22

There are several examples. I think the most prominent is for computing tensor products.
It holds that: $$\mathbb{Z}_a \otimes \mathbb{Z}_b = \mathbb{Z}_{gcd(a,b)}$$

Then you can use this identity to calculate tensors for all finitely generated abelian groups.

This might be especially interesting if you get into (co-)homology. At least in my eyes.

• That is a very good one.. Thank you! If you have another example, I will like to know.. Feb 18 at 13:56

Fact: If $$G$$ is a finite abelian group, then $$G \cong \mathrm{Hom}(G,\mathbb{C}^\times)$$.

Here, $$\mathbb{C}^\times$$ is the group of complex numbers under multiplication and the group law on the hom-set is pointwise multiplication.

The most straightforward way to prove this is to:

1. Prove it is true for finite cyclic groups;
2. Prove $$\mathrm{Hom}(H \times K,\mathbb{C}^\times) \cong \mathrm{Hom}(H,\mathbb{C}^\times) \times \mathrm{Hom}(K,\mathbb{C}^\times)$$ for any groups $$H,K$$; and then
3. Apply the classification of F-GAG.