Structure theorem for abelian groups

X]\times(k[X]/(X-1))\times (k[X]/(X-2))=k[X]\times\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}In summary, the presentation matrix A can be interpreted as the direct sum of the trivial group of just the identity plus the cyclic groups (t-1)^2(t-2), (t-1), and (t-2).
  • #1
Jim Kata
204
10
So say you have a presentation matrix A for a module, and you diagonalise it and you get something like diag[1,5] well you can interpret that as A can be broken down as the direct sum of 1+Z/5Z. That is the trivial group of just the identity plus cyclic 5. What if your module is defined over the ring of polynomials, and after you diagonalise your presentation matrix you get something like
diag[1,(t-1)^2(t-2)]? How do I interpret this? What is the cyclic group something like
(t-1)^2(t-2) would represent?
 
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  • #2
I assume you are working with k[X]. Your presentation matrix of M is

[tex]A=\left(\begin{array}{cc} 1 & 0\\ 0 & (X-1)^2(X-2)\\ \end{array}\right)[/tex]

Let the map corresponding to this matrix by [itex]\alpha:k[X]^2\rightarrow k[X]^2[/itex]. Then we know that

[tex]M\cong k[X]^2/\alpha k[X]^2[/tex]

Now, let {e,e'} be a basis corresponding to A, then we know that [itex]\alpha k[X][/itex] is generated by [itex]\{e,(X-1)^2(X-2)e'\}[/itex]

Thus

[tex]M\cong k[X]^2/\alpha k[X]^2=k[X]\times (k[X]/(X-1)^2(X-2))[/tex]
 

FAQ: Structure theorem for abelian groups

What is the structure theorem for abelian groups?

The structure theorem for abelian groups is a fundamental result in the field of abstract algebra. It states that every finite abelian group can be uniquely expressed as a direct product of cyclic groups. In other words, any abelian group can be broken down into simpler, cyclic subgroups.

Why is the structure theorem for abelian groups important?

The structure theorem for abelian groups is important because it provides a systematic way to understand and classify finite abelian groups. It also has many applications in other areas of mathematics, such as number theory and algebraic geometry.

How is the structure theorem for abelian groups proven?

The proof of the structure theorem for abelian groups involves using induction and the fundamental theorem of finitely generated abelian groups. It also relies on the fact that every finite cyclic group is isomorphic to a subgroup of the multiplicative group of complex numbers.

Are there any exceptions to the structure theorem for abelian groups?

Yes, there are a few exceptions to the structure theorem for abelian groups. The most notable one is the Klein four-group, which cannot be expressed as a direct product of cyclic groups. However, this is the only exception for finite abelian groups.

How is the structure theorem for abelian groups related to other theorems in mathematics?

The structure theorem for abelian groups is closely related to other theorems in mathematics, such as the classification of finitely generated modules over a principal ideal domain and the Chinese remainder theorem. It also has connections to the classification of finite abelian extensions in algebraic number theory.

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