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abelian group
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(Definition)
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Let  be a group. If for any  we have  , we say that the group is abelian (or commutative). Abelian groups are named after Niels Henrik Abel, but the word abelian is commonly written in lowercase.
Abelian groups are essentially the same thing as unitary
 -modules. 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:
Proof. Let  be a subgroup of the abelian group  . Since  for any  and any  we get  . That is,  is normal in  . 
Proof. Let  be a subgroup of  . Since  is abelian,  is normal and we can get the quotient group  whose elements are the equivalence classes for  if
 . The operation on the quotient group is given by
 . But
 , therefore the quotient group is also commutative. 
Here is another theorem concerning abelian groups:
Proof. 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. 
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"abelian group" is owned by yark. [ full author list (4) | owner history (5) ]
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(view preamble)
See Also: group, Klein 4-group, commutative semigroup, locally cyclic group, abelian groups of order  , fundamental theorem of finitely generated abelian groups, nonabelian group, commutative, metabelian group
| Other names: |
commutative group |
| Also defines: |
abelian, commutative |
| Keywords: |
group, abelian, commutative, quotient |
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Cross-references: function, operation, equivalence classes, quotient groups, normal, group
There are 212 references to this entry.
This is version 22 of abelian group, born on 2003-10-15, modified 2007-12-11.
Object id is 5107, canonical name is AbelianGroup2.
Accessed 29886 times total.
Classification:
| AMS MSC: | 20K99 (Group theory and generalizations :: Abelian groups :: Miscellaneous) |
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Pending Errata and Addenda
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