proof of fundamental theorem of finitely generated abelian groups
where . The numbers are uniquely determined as well as the number of ’s, which is the rank of an abelian group.
Proof. Let be an abelian group with generators. Then for a free group , is isomorphic to the quotient group . Now and contain a basis and satisfying for all . As , it suffices to show that is a direct sum of its cyclic subgroups .
It is clear that is generated by its subgroups . Assume that the zero element of can be written as a form . It follows that . As we write as a linear combination of that basis and using we get the equations
As every element can be represented uniquely as a linear combination of its free generators , we have for every and for every .
This means that every element belongs to , so . Therefore the zero element has a unique representation as a sum of the elements of the subgroup .
- 1 P. Paajanen: Ryhmäteoria. Lecture notes, Helsinki university, Finland (fall 2008)
|Title||proof of fundamental theorem of finitely generated abelian groups|
|Date of creation||2013-03-22 18:24:58|
|Last modified on||2013-03-22 18:24:58|
|Last modified by||puuhikki (9774)|