# Understanding rules on an Abelian group decomposition

In Aluffi's book Algebra, just from possibility of writing an abelian group G isomorphic to $$\langle g\rangle \oplus \ G/\langle g\rangle$$ it concludes that by induction $$G \cong \mathbb{Z}/d_1\mathbb{Z} \oplus \dots \mathbb{Z}/d_n\mathbb{Z}$$ with specific rules on $$d_i$$'s!

1- So why not $$G \cong \mathbb{Z}/p_1\mathbb{Z} \oplus \mathbb{Z}/p_1\mathbb{Z} \dots \mathbb{Z}/p_1\mathbb{Z} \oplus \mathbb{Z}/p_2\mathbb{Z} \dots \mathbb{Z}/p_2\mathbb{Z} \oplus \dots \dots \dots \mathbb{Z}/p_n\mathbb{Z}$$?

2- Why $$d_i|d_{i+1}$$? (a must/theorem or a choice/standard-rule?)

3- And if writing them as powers of primes so why following the below steps?

I'm assuming you're talking about finite abelian groups, otherwise one would need a free group $$\mathbb{Z}^r$$ in the decomposition.
1) We may not have necessarily have prime orders. Note $$ \cong \mathbb{Z}/d\mathbb{Z}$$ if $$g$$ has order $$d$$ in $$G$$. As an example, $$\mathbb{Z}/4\mathbb{Z} \not\cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$.
1. Elementary divisors and invariant factors are different things. The elementary divisors are a decomposition of the group into $$primary$$ ideals, that is power of prime ideals. One can determine these primary ideals given the invariant factors using the process described above.
• But $\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}$. A better example might be $\mathbb{Z}/4\mathbb{Z}$. – André 3000 Dec 2 '18 at 9:06