Direct groups

#matematik-groups

Definition: Direct product

Let be a family of groups. We define the direct product of these groups as the group given by the set where denotes a disjoint union, meaning we preserve the information about which group each element comes from. The operation of the direct product is defined componentwise as for all .

We want to preserve the information about which group each element comes from because we want to, given two elements, know which operation should be used on them. This definition allows for index sets of any size and cardinality. If the family of groups is of finite size we can see each element of the product as an -tuple, and if the index set is countably infinite we can see each element of the product as an infinite tuple, just like with regular set products.

Notation

If each group in a family is the same, so that for all , we denote the product as . If we denote the product as .