Confusion About Theorem for Finitely Generated Abelian Groups

Question

I was wondering about finitely generated abelian groups. Am I understanding this correctly? We may apply the structure theorem for finitely generated abelian groups in the following manner:

$\mathbb{Z}/12\mathbb{Z}\cong \mathbb{Z}/3\mathbb{Z}\oplus \mathbb{Z}/4\mathbb{Z}$

as well as

$\mathbb{Z}/10\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/5\mathbb{Z}$?

On top of this, I was wondering if we can claim in general that $\mathbb{Z}/n\mathbb{Z}$ is abelian under addition? Thanks.

Answer 1

In both examples, your decompositions are correct. However, the Fundamental Theorem of Finitely Generated Abelian Groups only provides that decompositions exist. It doesn't tell you how to decompose specific Abelian groups.

In the case of your second example, we use the fact that the prime factors of $10$ only appear with multiplicity one; i.e., $10 = 2 \cdot 5$. So the primary decomposition formulation of the Fundamental Theorem necessitates that $\mathbb{Z}/10\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z}$.

As $12 = 2^{2} \cdot 3$, the Fundamental Theorem doesn't tell us whether the direct factor of order $4$ for $\mathbb{Z}/12\mathbb{Z}$ is $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{V}_{4}$. One needs to look at the group more closely to make this determination.

On top of this, I was wondering if we can claim in general that Z/nZ is abelian under addition?

Yes- $\mathbb{Z}/n\mathbb{Z}$ is cyclic. Namely, $\mathbb{Z}/n\mathbb{Z} = \langle \overline{1} \rangle$.

Answer 2

Both results you quote are true by CRT. In general, if $(m,n)=1$, then $\Bbb Z_{mn}\cong\Bbb Z_m\times\Bbb Z_n$ (and conversely). This also follows from the fundamental theorem of finite abelian groups (FTFAG), which the theorem you mentioned generalizes.

Next $\Bbb Z_n:=\Bbb Z/n\Bbb Z$ is cyclic, hence abelian.