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All of the following ${p_{i},q_{i}}$are prime numbers, ${n,m,k}$ are pre-assigned integers. Consider the product of cyclic groups $\prod_{1}^{n} \mathbb Z_{p_{i}^{n_{i}}}$ then we asked the question:

How many subgroups of order $\prod_{1}^{n} p_{i}^{m_{i}}$ are there in the finite product of cyclic group?$(m_{i} \leqslant n_{i})$

For the initial problem, it's relatively easier,

How many subgroups of order $2^k$ in the product $\mathbb Z_{2^p} \times > \mathbb Z_{2^q}$? $(k \leqslant p \times q)$

It's not considered as a homework problem, I guess. Back ground My tutor once stated this problem as some simple application of Sylow Thm and found herself stuck there...

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See my answer to this question: mathoverflow.net/questions/78956/… –  Greg Martin Apr 2 '13 at 5:35
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Why not just consider one prime at a time? –  S. Carnahan Apr 2 '13 at 6:39
    
I hope the revised question communicates what you intended to ask. –  S. Carnahan Apr 2 '13 at 6:48
    
I might misused the headscript, I did correct it, and now it's clear. Mr. Martin's answer may solve my question greatly, but we discussed this afternoon and the problem is to deal with the different torsions arisen in Fund. Gen. Abelian group Thm. So I don't know if the attempt to reduce the problem to a single prime at a time will cause loss to generality. At least my tutor stated this problem twice and both in the same form. Reason why I tagged it representation problem is because I did remember a similar exercise related to it. –  Henr.L Apr 2 '13 at 10:41
    
FYI:And a more complicated version of the problem is to restrict ourselves to only one prime but with infinite product. –  Henr.L Apr 2 '13 at 10:41

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