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The direct product of two groups equals their direct sum. What is the difference in the case of infinite product and sum?

user3474383
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  • Do you know how the direct product and direct sum are defined in the infinite case? – Pedro Mar 28 '14 at 22:42
  • Well, I start suspecting that I'm missing something... – user3474383 Mar 28 '14 at 22:44
  • The direct product is simply the set of maps $f:I\to\bigcup_{i\in I} A_i$ with $f(i)\in A_i$ with component-wise multiplication and addition. The direct sum, on the other hand, is the set of maps $f:I\to\bigcup_{i\in I} A_i$ with $f(i)\in A_i$ and $f(i)=1_{A_i}$ for all but finitely many $i\in I$. That is, it is the set of finitely supported maps. – Pedro Mar 28 '14 at 22:53
  • It seems that you're trying to understand the funtamental group of the Hawaian earing... good luck! :) – user126154 Mar 29 '14 at 08:29

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The direct product of infinitely many groups is defined basically the same way it is for finitely many groups. It is the cartesian product of the sets with componentwise operations. The direct sum of infinitely many groups is a proper subset of the direct product, defined as the collection of those elements for which all but finitely many components are the identity.

Seth
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