Tensor product of abelian groups

As tensor product of modules
Suppose $$G$$ and $$H$$ are abelian groups (possibly equal, possibly distinct). Their tensor product as abelian groups, denoted $$G \otimes_{\mathbb{Z}} H$$ or simply as $$G \otimes H$$, is defined as their tensor product as modules over the ring of integers $$\mathbb{Z}$$.

Note that in case $$G,H$$ are abelian groups but are also being thought of as modules over some other ring (for instance, as vector spaces over a field) then the notation $$G \otimes H$$ is ambiguous as it is not clear what ring the tensor product is being carried out over. Generally, if the vector space structure is salient in the context, the tensor product is as vector spaces. Thus, in such circumstances, it is important to write $$\otimes_{\mathbb{Z}}$$ when talking about the tensor product of abelian groups.

Explicit definition
Suppose $$G$$ and $$H$$ are abelian groups (possibly equal, possibly distinct). Their tensor product as abelian groups, denoted $$G \otimes_{\mathbb{Z}} H$$ or simply as $$G \otimes H$$, is defined as the quotient of the free abelian group on the set of all symbols $$\{ g \otimes h \mid g \in G, h \in H \}$$ by the following relations:


 * $$(g_1 + g_2) \otimes h = (g_1 \otimes h) + (g_2 \otimes h)$$ for all $$g_1,g_2 \in G, h \in H$$
 * $$g \otimes (h_1 + h_2) = (g \otimes h_1) + (g \otimes h_2)$$ for all $$g \in G, h_1,h_2 \in H$$

In other words, the mapping $$(g,h) \mapsto g \otimes h$$ is a bihomomorphism.

Note: For the general definition of tensor product of modules, we need to additionally put conditions saying that ring scalars can be pulled out of tensor products. This, however, is not necessary for the definition with abelian groups as it follows from the additive conditions listed above. That is because every element of $$\mathbb{Z}$$ can be written as sums and differences of 1s.

Definition in terms of tensor product of groups
The tensor product of abelian groups $$G,H$$ agrees with the tensor product of groups if we assume both groups to act trivially on each other. Note that trivial mutual actions form a compatible pair of actions, so the definition applies.