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How to tell if a product of cyclic groups $C_{n_1}\times C_{n_2} \times ...\times C_{n_k}$, where $n_1+n_2+...+n_k=N$, is isomorphic to another product of cyclic groups $C_{m_1}\times C_{m_2} \times ...\times C_{m_j}$, where $m_1+m_2+...m_j=M$, given that $N=M$?

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  • $\begingroup$ You probably mean $n_1 \cdot n_2 \cdots n_k = N$. $\endgroup$ – lhf Aug 3 '17 at 18:37
  • $\begingroup$ I wonder whether we can do this without factoring, just using gcd and lcm. It works $k=2$. $\endgroup$ – lhf Aug 3 '17 at 18:39
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Write each of $n_1, n_2,\dots, n_k$ and each of $m_1,m_2,\dots, m_j$ as a power of prime powers to obtain the list of elementary divisors of each product and compare these lists. The groups are isomorphic if and only if these lists are equal.

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  • $\begingroup$ For two different collections of elementary divisors, how can we prove that the two Abelian groups derived from them respectively are not isomorphic? $\endgroup$ – Daniel Li Aug 3 '17 at 19:05
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    $\begingroup$ That is Kronecker's theorem, a.k.a. fundamental theorem of finite abelian groups. Do you want to prove it? $\endgroup$ – Bernard Aug 3 '17 at 20:13
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Every finite abelian group is a product of cyclic groups of prime power order. In fact, this decomposition is unique up to the multiset of prime powers.

So, in your case, you would just have to factor each $n_i$ and $m_i$ into their prime power factors, and then compare the (multi-)sets of all prime power factors of all the $n_i$s with that of all the $m_i$s.

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