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:
“I esteem it the happiness of this country that its settlers, whilst they were exploring their granted and natural rights and determining the power of the magistrate, were united by personal affection. Members of a church before whose searching covenant all rank was abolished, they stood in awe of each other, as religious men.”
—Ralph Waldo Emerson (1803–1882)
“Laughing at someone else is an excellent way of learning how to laugh at oneself; and questioning what seem to be the absurd beliefs of another group is a good way of recognizing the potential absurdity of many of one’s own cherished beliefs.”
—Gore Vidal (b. 1925)