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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.

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Going from the second to the first involves factorization of integers, which is believed to a computationally difficult problem (the security of your bank account probably depends on this). But it is computationally easy to go from (1) to (2).

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.

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It all depends on what you mean by "immediately".

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.

But it may be worth repeating that if the group is given by generators and relations, then it's usually feasible to calculate the invariant factors,even when these are too big for it to be feasible to factorise these into primes, since the invariant factors can be found by a Euclidean algorithm based procedure.

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