proof of fundamental theorem of finitely generated abelian groups

Every finitely generatedMathworldPlanetmathPlanetmath abelian groupMathworldPlanetmath A is a direct sumPlanetmathPlanetmathPlanetmath of its cyclic subgroups, i.e.


where  1<m1m2mk. The numbers mi are uniquely determined as well as the number of ’s, which is the rank of an abelian group.

Proof. Let G be an abelian group with n generatorsPlanetmathPlanetmathPlanetmath. Then for a free groupMathworldPlanetmath Fn, G is isomorphicPlanetmathPlanetmathPlanetmath to the quotient groupMathworldPlanetmath Fn/A. Now Fn and A contain a basis f1,,fn and a1,,ak satisfying ai=mifi for all 1ik. As GFn/A, it suffices to show that Fn/A is a direct sum of its cyclic subgroups f1+A.

It is clear that Fn/A is generated by its subgroupsMathworldPlanetmathPlanetmath fi+A. Assume that the zero elementMathworldPlanetmath of Fn/A can be written as a form A=l1f1++lnfn+A. It follows that l1f1++lnfn=aA. As we write a as a linear combinationMathworldPlanetmath of that basis and using ai=mifi we get the equations


As every element can be represented uniquely as a linear combination of its free generators f1, we have li=simi for every 1ik and lj=0 for every k<jn.

This means that every element lifi belongs to A, so  lifi+A=A. Therefore the zero element has a unique representation as a sum of the elements of the subgroup fi+A.


  • 1 P. Paajanen: Ryhmäteoria.  Lecture notes, Helsinki university, Finland (fall 2008)
Title proof of fundamental theorem of finitely generated abelian groups
Canonical name ProofOfFundamentalTheoremOfFinitelyGeneratedAbelianGroups
Date of creation 2013-03-22 18:24:58
Last modified on 2013-03-22 18:24:58
Owner puuhikki (9774)
Last modified by puuhikki (9774)
Numerical id 7
Author puuhikki (9774)
Entry type Proof
Classification msc 20K25