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What is the difference between direct sum and restricted product of abelian group?

I recently heard that direct limit of direct product is not a direct sum but a restricted product.

Until now, I thought both are the same set, that is, $\bigoplus_{i\in I} M_i=\prod'_{i \in I}M_i$ where $M_i$ is $0$ except for finitely many $I$(infinite set). What is the difference between $\bigoplus_{i\in I} M_i$ and $\prod'_{i \in I}M_i$ are there inclusion between them ?

Thank you in advance.

P.S.

Sorry for typo in my title. What I'm asking is difference between direct sum and restricted product. Notice $\prod'$ is different from $\prod$.

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  • $\begingroup$ Elements of a direct sum (or restricted direct product) must have finitely many nonzero coordinates. No such restriction is placed for elements of a direct product. $\endgroup$ Commented May 21 at 15:14
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    $\begingroup$ What definition of restricted product do you use? For example, in the following definition you can see the difference better: encyclopediaofmath.org/wiki/Restricted_direct_product $\endgroup$ Commented May 21 at 17:28

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There are many types of products you can do. This only applies to products of infinitely many (normal) subgroups, but also not just to abelian groups. There are different names for these depending on whom you talk to, so I will try to be careful.

Let $I$ be an indexing set and let $\{G_i\mid i\in I\}$ be a collection of groups. The direct sum (also called direct product, or restricted direct product by group theorists) of the $G_i$ is the set of all tuples $$ (g_i)_{i\in I}$$ where all but finitely many of the $g_i$ are the identity.

The direct product (also called Cartesian product, or unrestricted direct product) of the $G_i$ is the set of all tuples $$ (g_i)_{i\in I}$$ where there is now no condition on the $g_i$ being the identity. In between them is another type of direct product: if $\kappa$ is some infinite cardinal, we can form the set of all tuples where the number of non-trivial $g_i$ has cardinality less than $\kappa$.

If $\kappa=\aleph_0$ we obtain the restricted direct product and if $\kappa>|I|$ we obtain the unrestricted direct product.

How exactly this collection nests depends on your axiomatic framework with regards infinite cardinals (particularly the axiom of choice), but the restricted and unrestricted direct products always exist, and the unrestrcted one clearly contains the restricted one.

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  • $\begingroup$ Sorry for my typo, but I want to know the difference between 'direct sum' (not direct product in the way we restrict the elements are $0$ except for finitely many $i$)and 'restricted product', both I thought is the same. $\endgroup$ Commented May 21 at 15:38
  • $\begingroup$ @Pont Why do you think they are different? Providing the sources for what you have been reading might help. $\endgroup$ Commented May 21 at 15:52
  • $\begingroup$ I meat last comment in answer of Christian Wuthrich. What do you think ? mathoverflow.net/questions/471560/… $\endgroup$ Commented May 21 at 16:00
  • $\begingroup$ @Pont I don't know for sure, but if forced to guess: the comment says limit rather than colomit, so direct product is needed rather than sum (limit = inverse limit. Direct limit = colimit.) The restricted product might mean one of the intermediate products that I mention, in between the sum and product. $\endgroup$ Commented May 21 at 19:22

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