what is the difference between product and sum of groups?

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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?

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1 Answer

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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.

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