abelian groups of order 120
Here we present an application of the fundamental theorem of finitely generated abelian groups.
Example (Abelian groups
of order 120):
Let G be an abelian group of order n=120. Since the group is
finite it is obviously finitely generated, so we can apply the
theorem. There exist n1,n2,…,ns with
| G≅ℤ/n1ℤ⊕ℤ/n2ℤ⊕…⊕ℤ/nsℤ |
| ∀i,ni≥2;ni+1∣nifor 1≤i≤s-1 |
Notice that in the case of a finite group, r,
as in the statement of the theorem, must be equal to 0. We have
| n=120=23⋅3⋅5=s∏i=1ni=n1⋅n2⋅…⋅ns |
and by the divisibility properties of ni we must have that
every prime divisor of n must divide n1. Thus the
possibilities for n1 are the following
| 2⋅3⋅5,22⋅3⋅5,23⋅3⋅5 |
If n1=23⋅3⋅5=120 then s=1. In the case that n1=22⋅3⋅5 then n2=2 and s=2. It remains to analyze the case n1=2⋅3⋅5. 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 isomorphism) one of the following:
| ℤ/120ℤ,ℤ/60ℤ⊕ℤ/2ℤ,ℤ/30ℤ⊕ℤ/2ℤ⊕ℤ/2ℤ |
Also notice that they are all non-isomorphic. This is because
| ℤ/(n⋅m)ℤ≅ℤ/nℤ⊕ℤ/mℤ⇔gcd(n,m)=1 |
which is due to the
Chinese Remainder theorem.
| 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 |