rational rank of a group

In the following, G is an abelian groupMathworldPlanetmath.

Definition 1.

The group QZG is called the divisible hull of G.

It is a -vector spaceMathworldPlanetmath such that the scalar -multiplication of G is extended to .

Definition 2.

The elements g1,g2,,grG are called rationally independent if they are linearly independentMathworldPlanetmath over Z, i.e. for all n1,,nrZ:

Definition 3.

The dimensionPlanetmathPlanetmath of QZG over Q is called the rational rank of G.

We denote the rational rank of G by r(G).
r(×)=2 because (×)=()×()=×.

  • If H is a subgroupMathworldPlanetmathPlanetmath of G then we have:


    It results from the fact that 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