# Rank of an Abelian group

The **rank of an Abelian group** is a term from the mathematical branch of algebra . It is a measure of the size of an Abelian group .

## definition

For an Abelian group, the following numbers match:

- 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 characteristics

- The rank of for a natural number is equal ; more generally, 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 to short exact sequences : is

- is an exact sequence of Abelian groups, then the rank of is equal to the sum of the ranks of and .