Encyclopedia > Free abelian group

  Article Content

Free abelian group

In abstract algebra, the free abelian group on a set X may be constructed as the abelian group of functions on X, taking integer values that are almost all zero. One can verify directly that this has the appropriate universal property in relation to arbitrary functions on X with values in some abelian group A: namely unique extension to a homomorphism of the free group.

When X is finite of cardinality n the free abelian group on X is the same up to isomorphism as the product of n copies of the infinite cyclic group. This breaks down for infinite X, though.

This construction is a special case of the construction of free modules[?].



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Leupp, Arizona

... the census of 2000, there are 970 people, 228 households, and 206 families residing in the town. The population density is 28.5/km² (73.9/mi²). There are 277 ...