weak product

For a collection of groups ${G_{i}}_{i \in I}$ , the weak product is the group $G$ whose elements are $(g_{i})_{i \in I}$ such that $g_{i} \in G_{i}$ and $g_{i} = e$ (the identity element in that group) for almost all $i$ and the group multiplication is componentwise.

Note: Depending on what properties you want from the product, you can consider $g_{i} = e$ for all but finitely entries.