abelian groups of order
Here we present an application of the fundamental theorem of finitely generated abelian groups.
Example (Abelian groups of order ):
Let be an abelian group of order . Since the group is finite it is obviously finitely generated, so we can apply the theorem. There exist with
Notice that in the case of a finite group, , as in the statement of the theorem, must be equal to . We have
If then . In the case that then and . It remains to analyze the case . Now the only possibility for is and as well.
Hence if is an abelian group of order it must be (up to isomorphism) one of the following:
Also notice that they are all non-isomorphic. This is because
which is due to the Chinese Remainder theorem.
|Title||abelian groups of order|
|Date of creation||2013-03-22 13:54:17|
|Last modified on||2013-03-22 13:54:17|
|Last modified by||alozano (2414)|