Finitely generated abelian groups

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In summary, the book states that a quotient group of a finitely generated abelian group by certain components is isomorphic to a product of cyclic groups. The explanation for this statement is not provided, but it can be proven using a surjective homomorphism and the first isomorphism theorem.
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ehrenfest
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[SOLVED] finitely generated abelian groups

Homework Statement


My book states that
[tex] (\mathbb{Z} \times\mathbb{Z} \times\cdots \times \mathbb{Z})/(d_1\mathbb{Z} \times d_2\mathbb{Z} \times \cdots d_s\mathbb{Z} \times {0} \times \cdots \times {0}) [/tex]
is isomorphic to
[tex] \mathbb{Z}_{d_1} \times\cdots \times \mathbb{Z}_{d_s} \times \mathbb{Z} \times\cdots \times \mathbb{Z}[/tex]

with absolutely no explanation of any sort. I don't know why this is so obvious to everyone because it is NOT TRUE that you can just mod out things by there components as you can clearly see from the fact that: (Z_4 \times Z_6)/<(2,3)> is not isomorphic to Z_2 \times Z_3.
I have absolutely no idea what thought process went into that statement above and why they think they can just mod things out by there components when that is just wrong.

Homework Equations





The Attempt at a Solution

 
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  • #2
Modding out by <(2,3)> is not quite like what they're doing.

Can you think of a surjective homomorphism from RxS onto (R/I) x (S/J)? What's its kernel?
 
  • #3
Define a homomorphism \phi: [tex] (\mathbb{Z} \times\mathbb{Z} \times\cdots \times \mathbb{Z}) \to \mathbb{Z}_{d_1} \times\cdots \times \mathbb{Z}_{d_s} \times \mathbb{Z} \times\cdots \times \mathbb{Z}[/tex]

as follows. For each of the components 1 \leq i \leq s, take that coordinate to itself mod d_i. For the rest of the cosets take that coordinate to itself. Then the kernel is obviously what we are modding out by. Then apply the first isomorphism theorem. Wow.
 
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Related to Finitely generated abelian groups

1. What is a finitely generated abelian group?

A finitely generated abelian group is a mathematical structure consisting of a set of elements and a binary operation that satisfies the properties of closure, associativity, identity, and inverse. Additionally, the group must be generated by a finite number of elements and must satisfy the commutative property, meaning that the order in which the elements are combined does not affect the result.

2. How do you determine the order of a finitely generated abelian group?

The order of a finitely generated abelian group is equal to the number of elements in the group. It can be determined by counting the total number of elements in the set or by taking the product of the orders of each of the generators.

3. What are some examples of finitely generated abelian groups?

Some examples of finitely generated abelian groups include the integers (under addition), the cyclic groups, and the direct product of cyclic groups. Other examples include finite groups such as the Klein four-group and the dihedral groups.

4. What is the significance of finitely generated abelian groups in mathematics?

Finitely generated abelian groups have many applications in mathematics, including in algebra, number theory, and topology. They also play a crucial role in the classification of groups and in understanding the structure of more complex groups.

5. What are some properties of subgroups of finitely generated abelian groups?

Subgroups of finitely generated abelian groups are themselves finitely generated and abelian. Additionally, the order of a subgroup must divide the order of the original group. Subgroups of finitely generated abelian groups also have a unique decomposition into cyclic groups.

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