Abelian groups are essentially the same thing as unitary -modules (http://planetmath.org/Module). In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using additive notation.
Some of the basic properties of abelian groups are as follows:
Any subgroup (http://planetmath.org/Subgroup) of an abelian group is normal.
Let be a subgroup of the abelian group . Since for any and any we get . That is, is normal in . ∎
Quotient groups of abelian groups are also abelian.
Here is another theorem concerning abelian groups:
If defined by is a homomorphism (http://planetmath.org/GroupHomomorphism), then is abelian.
If such a function were a homomorphism, we would have
that is, . Left-multiplying by and right-multiplying by we are led to and thus the group is abelian. ∎
|Date of creation||2013-03-22 14:01:55|
|Last modified on||2013-03-22 14:01:55|
|Last modified by||yark (2760)|