abelian groups of order 120

Here we present an application of the fundamental theorem of finitely generated abelian groups.

Example (Abelian groupsMathworldPlanetmath of order 120):

Let G be an abelian group of order n=120. Since the group is finite it is obviously finitely generatedMathworldPlanetmathPlanetmathPlanetmath, so we can apply the theorem. There exist n1,n2,,ns with

i,ni2;ni+1nifor 1is-1

Notice that in the case of a finite groupMathworldPlanetmath, r, as in the statement of the theorem, must be equal to 0. We have


and by the divisibility properties of ni we must have that every prime divisorMathworldPlanetmathPlanetmath of n must divide n1. Thus the possibilities for n1 are the following


If n1=2335=120 then s=1. In the case that n1=2235 then n2=2 and s=2. It remains to analyze the case n1=235. Now the only possibility for n2 is 2 and n3=2 as well.

Hence if G is an abelian group of order 120 it must be (up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) one of the following:


Also notice that they are all non-isomorphic. This is because


which is due to the Chinese Remainder theoremMathworldPlanetmathPlanetmathPlanetmath.

Title abelian groups of order 120
Canonical name AbelianGroupsOfOrder120
Date of creation 2013-03-22 13:54:17
Last modified on 2013-03-22 13:54:17
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Example
Classification msc 20E34
Related topic FundamentalTheoremOfFinitelyGeneratedAbelianGroups
Related topic AbelianGroup2