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)
“A man is the prisoner of his power. A topical memory makes him an almanac; a talent for debate, disputant; skill to get money makes him a miser, that is, a beggar. Culture reduces these inflammations by invoking the aid of other powers against the dominant talent, and by appealing to the rank of powers. It watches success.”
—Ralph Waldo Emerson (1803–1882)
“No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.”
—bell hooks (b. c. 1955)