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 . ${\displaystyle G}$ ${\displaystyle G'}$ ${\displaystyle G''}$