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

G/n1/n2/ns
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

n=120=2335=i=1sni=n1n2ns

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

235,2235,2335

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:

/120,/60/2,/30/2/2

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

/(nm)/n/mgcd(n,m)=1

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