Hi, let S be any set and let ##Z\{S\}## be the free group on ##S##, i.e., ##Z\{S\}## is(adsbygoogle = window.adsbygoogle || []).push({});

the collection of all functions of finite support on ##S##. I am trying to show

that for an Abelian group ##G## , we have that :

## \mathbb Z\{S\}\otimes G \sim |S|G = \bigoplus_{ s \in S} G ##, i.e., the direct sum of ##|S|## copies of ##G## , where ## \otimes ## is the tensor product of Abelian groups.

I know that for ## \mathbb Z ## the integers , then ##\mathbb Z \otimes G \sim G ##( tensor over the integers ) by, e.g., the map ##(z,g)\rightarrow zg =g+g+...+g## ##z## times. I know two that any

basis for ## \mathbb Z\{S\} ## has cardinality that of ##S##.

But I don't see why ## \mathbb Z\{S\}\otimes G = \bigoplus_{ s \in S} G## , the direct sum of ##|S|## copies of ##G##.

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# Tensors of Free Groups and Abelian groups

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