External Direct Products
External direct products are basically n-tuples of elements from different groups.
Ifis a collection of groups, the external direct productis the set of all n – tuples with the ith component is an element ofWe can writeand define the product of elements ofasThe productis done using the group operation of
The direct product of any number of groups is itself a group. If the order ofis m-i then the order ofis
An obvious example of a direct product isEach component ofis a real number.is a group with the group operation being addition, sois a group with the group operation being componentwise addition.
is a group with components first component one offrom and the second component one offrom
If all theare abelian, then so is
If theare cyclic of orderrespectively, and none of thehave any common factors, thenis cyclic of order