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I am trying to understand the relation between the direct sum and direct product of all the groups from the set

$$\{S_{N_1} \wr D_{m_1}, \ldots, S_{N_i} \wr D_{m_i}, \ldots, S_{N_t} \wr D_{m_t}\}$$

where $i, t, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of order $2 m_i$.

It looks like there is a subgroup in the direct product which is isomorphic to the direct sum. But I am not able to tell more than that.

Anyone any help?

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    $\begingroup$ The direct product and direct sum of finitely many groups (assuming you mean what I think you mean by "direct sum") are the same. $\endgroup$ – Qiaochu Yuan Dec 18 '15 at 18:13
  • $\begingroup$ @QiaochuYuan, I am using this definition for direct sum. en.wikipedia.org/wiki/Direct_sum_of_groups $\endgroup$ – Omar Shehab Dec 18 '15 at 18:48

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