Let be any nonempty set and let be a group for each . The direct product of the groups is the set (the Cartesian product of the ) with a binary operation defined as follows: if and are elements of , then their product is defined . I.e., the product is defined componentwise.
- Show that this operation is well defined and associative.
- Show that the element is an identity in .
- Show that the element is the inverse of .
Conclude that is a group.
- Elements in a direct product are uniquely represented, so well-definedness is not an issue. Suppose , , and are in . Then , so that the product in is associative.
- We have ; likewise . Thus is an identity in under componentwise multiplication.
- We have ; likewise . Thus .
Thus is a group under componentwise multiplication.