Example
For example, let G be the group that is the direct sum of two copies of the infinite cyclic group . Symbolically,
- .
One basis for this group is {(1,0),(0,1)}. If we say and, then we can write the element (4,3) as
- . Where 'multiplication' is defined in following way: .
In this basis, there is no other way to write (4,3), but if we choose our basis to be {(1,0),(1,1)}, where and, then we can write (4,3) as
- .
Unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do. (For instance, any group having periodic elements is not a free abelian group because any element can be expressed in an infinite number of ways simply by putting in an arbitrary number of cycles constructed from a periodic element.) The trivial abelian group {0} is also considered to be free abelian, with basis the empty set.
Every lattice forms a finitely-generated free abelian group.
Read more about this topic: Free Abelian Group
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)