Rank of An Abelian Group

Rank Of An Abelian Group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.

The term rank has a different meaning in the context of elementary abelian groups.

Read more about Rank Of An Abelian Group:  Definition, Properties, Groups of Higher Rank, Generalization

Famous quotes containing the words rank and/or group:

    Oxford is a little aristocracy in itself, numerous and dignified enough to rank with other estates in the realm; and where fame and secular promotion are to be had for study, and in a direction which has the unanimous respect of all cultivated nations.
    Ralph Waldo Emerson (1803–1882)

    The government of the United States at present is a foster-child of the special interests. It is not allowed to have a voice of its own. It is told at every move, “Don’t do that, You will interfere with our prosperity.” And when we ask: “where is our prosperity lodged?” a certain group of gentlemen say, “With us.”
    Woodrow Wilson (1856–1924)