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 of, rank and/or group:

    West of this place, down in the neighbor bottom,
    The rank of osiers by the murmuring stream
    Left on your right hand brings you to the place.
    William Shakespeare (1564–1616)

    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)

    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)