# comparison of two types of decomposition of finite abelian groups

If $G$ is a finite abelian group then $G$ has a decomposition into two types:

(1) one is direct product of cyclic groups of some prime power order (may be with repitition)

(2) other is direct product of cyclic groups where order of one component divides order of next one (invariant factors).

In the comparison of these two factorisations, I came across following natural question, which is usually not raised or discussed in classrooms or in books.

Q. What information about $G$ can be immediately given from one factorization which is not immediate from other factorization?

For example, if we know factorization $\mathbb{Z}_{d_1}\times \cdots\times\mathbb{Z}_{d_r}$ with $d_i|d_{i+1}$, we can say what is exponent, whether group is cyclic. I don't know beyond this, for what purpose one fattorisation is useful than other.

So one answer is that (1) provides you immediately with a prime factorization of $|G|$ (and makes it easy to compute Sylow subgroups), whereas (2) does not.
We can pass between the two factorisations by simple arithmetic after all: with this caveat, provided we find it feasible to factorise the $d_i$.
To see this, just use the Chinese Remainder Thm, that $\mathbb{Z}_{mn}=\mathbb{Z}_m\oplus\mathbb{Z}_n$ when $m,n$ are coprime, to collect together the appropriate cycles of prime-power order, or to split the invariant cycles.