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Every finite abelian group $G$ can be uniquely written as $$\mathbb{Z}/{d_1\mathbb{Z}} \times \mathbb{Z}/{d_2\mathbb{Z}} \times \cdots \times \mathbb{Z}/{d_r\mathbb{Z}},$$ where $d_i$ divides $d_{i+1}$ and $\prod_{i}d_i=|G|$.

Also, $G$ can be uniquely written as direct product of Sylow-$p$ subgroups, or even finer, $$\mathbb{Z}/p_1^{n_1}\mathbb{Z} \times \mathbb{Z}/p_2^{n_2}\mathbb{Z} \times \cdots \times \mathbb{Z}/p_k^{n_k}\mathbb{Z},$$ ($p_i$'s are primes that are non-necessarily distinct, $n_i\geq 1$, and $\prod_i p_i^{n_i}=|G|$).

To study some problems on abelian groups, one form works better than the other. For example, the decomposition into Sylow subgroups is helpful to understand the irreducible representations of group $G$.

Question Can one give some other examples where one form of decomposition is better than the other? (especially, the first decomposition is helpful than the other?)

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    $\begingroup$ Perhaps Dummit and Foote's Abstract Algebra will help you. Look at page 158, it has both and a couple of useful examples! $\endgroup$ – Bill Moustakas Jul 2 '15 at 9:11
  • $\begingroup$ Just a trivial comment, if you want uniqueness in the first decomposition, take $d_{i} > 1$. $\endgroup$ – Andreas Caranti Jul 2 '15 at 12:35

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