# rational rank of a group

In the following, $G$ is an abelian group.

###### Definition 1.

The group $\mathbb{Q}\otimes_{\mathbb{Z}}G$ is called the divisible hull of $G$.

It is a $\mathbb{Q}$-vector space such that the scalar $\mathbb{Z}$-multiplication of $G$ is extended to $\mathbb{Q}$.

###### Definition 2.

The elements $g_{1},g_{2},...,g_{r}\in G$ are called rationally independent if they are linearly independent over $\mathbb{Z}$, i.e. for all $n_{1},...,n_{r}\in Z$:

 $n_{1}g_{1}+...+n_{r}g_{r}=0\Rightarrow n_{1}=...=n_{r}=0.$
###### Definition 3.

The dimension of $\mathbb{Q}\otimes_{\mathbb{Z}}G$ over $\mathbb{Q}$ is called the rational rank of $G$.

We denote the rational rank of $G$ by $r(G)$.
Example:
$r(\mathbb{Z}\times\mathbb{Z})=2$ because $\mathbb{Q}\otimes_{\mathbb{Z}}(\mathbb{Z}\times\mathbb{Z})=(\mathbb{Q}\otimes_% {\mathbb{Z}}\mathbb{Z})\times(\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Z})=% \mathbb{Q}\times\mathbb{Q}$.
Properties:

• If $H$ is a subgroup of $G$ then we have:

 $r(G)=r(H)+r(G/H).$

It results from the fact that ${}_{\mathbb{Z}}\mathbb{Q}$ is a flat module.

• The rational rank of the group $G$ can be defined as the least upper bound (finite or infinite) of the cardinals $r$ such that there exist $r$ rationally independent elements in $G$.

 Title rational rank of a group Canonical name RationalRankOfAGroup Date of creation 2013-03-22 16:53:25 Last modified on 2013-03-22 16:53:25 Owner polarbear (3475) Last modified by polarbear (3475) Numerical id 9 Author polarbear (3475) Entry type Definition Classification msc 06F20 Classification msc 20K20 Classification msc 20K15 Classification msc 20K99 Defines rationally independent Defines rational rank Defines divisible hull