# 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: ${\ displaystyle G}$

• the cardinality of a maximum -linearly independent subset${\ displaystyle \ mathbb {Z}}$
• the dimension of the vector space (see tensor product ).${\ displaystyle \ mathbb {Q}}$${\ displaystyle G \ otimes \ mathbb {Q}}$

This number is called the rank of . ${\ displaystyle G}$

## 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 .${\ displaystyle \ mathbb {Z} ^ {n}}$ ${\ displaystyle n}$${\ displaystyle n}$ ${\ displaystyle \ mathbb {Z} ^ {(I)}}$${\ displaystyle I}$${\ displaystyle I}$
• The group has rank n .${\ displaystyle \ mathbb {Q} ^ {n}}$
• An Abelian group is a torsion group if and only if its rank is 0.
• The rank is additive to short exact sequences : is
${\ displaystyle 0 \ longrightarrow G '\ longrightarrow G \ longrightarrow G' '\ longrightarrow 0}$
is an exact sequence of Abelian groups, then the rank of is equal to the sum of the ranks of and .${\ displaystyle G}$${\ displaystyle G '}$${\ displaystyle G ''}$