Finitely-generated Abelian Group

Finitely-generated Abelian Group

In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form

x = n1x1 + n2x2 + ... + nsxs

with integers n1,...,ns. In this case, we say that the set {x1,...,xs} is a generating set of G or that x1,...,xs generate G.

Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.

Read more about Finitely-generated Abelian Group:  Examples, Classification, Corollaries, Non-finitely Generated Abelian Groups

Famous quotes containing the word group:

    The virtue of dress rehearsals is that they are a free show for a select group of artists and friends of the author, and where for one unique evening the audience is almost expurgated of idiots.
    Alfred Jarry (1873–1907)