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:
“Only what is rare is valuable.
Let no one dare to call another mad who is not himself willing to rank in the same class for every perversion and fault of judgment. Let no one dare aid in punishing another as criminal who is not willing to suffer the penalty due to his own offenses.”
—Margaret Fuller (1810–1850)
“A private should preserve a respectful attitude toward his superiors, and should seldom or never proceed so far as to offer suggestions to his general in the field. If the battle is not being conducted to suit him, it is better for him to resign. By the etiquette of war, it is permitted to none below the rank of newspaper correspondent to dictate to the general in the field.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The rank is but the guinea stamp—
The man’s the gowd for a’ that!”
—Robert Burns (1759–1796)