Rank of an abelian group

The rank of an abelian group is a concept from the mathematical field of algebra. It is a measure for the size of an abelian group.

Definition

For an abelian group agree the following numbers agree:

• The cardinality of a maximum - linearly independent subset
• The dimension of the vector space (see tensor product ).

This number is called the rank of.

Examples and properties

• The rank of a natural number is the same; general the rank of the free abelian group on a set is equal to the cardinality of.
• The group has rank n
• An abelian group is a torsion group if and only if its rank is 0.
• The rank is additive on short exact sequences: Is

• Group Theory