Free Abelian Group - Rank

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:

    Lady Hodmarsh and the duchess immediately assumed the clinging affability that persons of rank assume with their inferiors in order to show them that they are not in the least conscious of any difference in station between them.
    W. Somerset Maugham (1874–1965)

    The office of the prince and that of the writer are defined and assigned as follows: the nobleman gives rank to the written work, the writer provides food for the prince.
    Franz Grillparzer (1791–1872)

    If we were left solely to the wordy wit of legislators in Congress for our guidance, uncorrected by the seasonable experience and the effectual complaints of the people, America would not long retain her rank among the nations.
    Henry David Thoreau (1817–1862)