Every finitely generated free abelian group is isomorphic to for some natural number n called the rank of the free abelian group. In general, a free abelian group F has many different bases, but all bases have the same cardinality, and this cardinality is called the rank of F. This rank of free abelian groups can be used to define the rank of all other abelian groups: see rank of an abelian group. The relationships between different bases can be interesting; for example, the different possibilities for choosing a basis for the free abelian group of rank two is reviewed in the article on the fundamental pair of periods.
Read more about this topic: Free Abelian Group
Famous quotes containing the word rank:
“... his rank penetrated them as though it had been an odour.”
—George Eliot [Mary Ann (or Marian)
“In a famous Middletown study of Muncie, Indiana, in 1924, mothers were asked to rank the qualities they most desire in their children. At the top of the list were conformity and strict obedience. More than fifty years later, when the Middletown survey was replicated, mothers placed autonomy and independence first. The healthiest parenting probably promotes a balance of these qualities in children.”
—Richard Louv (20th century)
“Its whether will ye be a rank robber’s wife,
Or will ye die by my wee pen knife?
Its I’ll not be a rank robber’s wife,
But I’ll rather die by your wee pen knife.
He ‘s killed this may and he ‘s laid her by,
For to bear the red rose company.”
—Unknown. Babylon; or, The Bonnie Banks o’ Fordie (l. 9–14)