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All of the following ${p_{i},q_{i},k_{i}}are prime numbers, ${n_{i},m_{i}}$ 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...

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...

All of the following $p_{i}$ $q$ and $k_{i}$ are prime number, $n_{i} ,m_{i}$ 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

All of the following ${p_{i},q_{i},k_{i}}are prime numbers, ${n_{i},m_{i}}$ 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...

Suppose $\{p_i\}$ are prime numbers, and $\{n_i ,m_i\}$ are pre-assigned integers. Consider the product of cyclic groups $G = \prod_{i=1}^{n} (\mathbb Z/p_{i}^{n_{i}}\mathbb Z)$.

How many subgroups of order $\prod_{i=1}^{n} p_{i}^{m_{i}}$ are there in $G$ (where $m_i \leqslant n_i$)?

Here is a problem that is relatively easier:

How many subgroups of order $2^k$ are there in the product $\mathbb Z/2^p\mathbb Z \times > \mathbb Z/2^q \mathbb Z$ (where $k \leqslant p + q$)?

All of the following $p_{i}$ $q$ and $k_{i}$ are prime number, $n_{i} ,m_{i}$ 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