Subgroup of Finitely Generated Abelian Group

In summary, the conversation discusses how to prove that any subgroup of a finitely generated abelian group is also finitely generated. One approach is to use induction, while another suggestion is to prove a general result that states if there exists a finitely generated subgroup with a finitely generated quotient group, then the original group is also finitely generated. The latter approach is then successfully used to prove the initial statement.
  • #1
jumpr
5
0

Homework Statement


Prove that any subgroup of a finitely generated abelian group is finitely generated.

Homework Equations




The Attempt at a Solution


I've attempted a proof by induction on the number of generators. The case n=1 corresponds to a cyclic group, and any subgroup of a cyclic group is cyclic, and so generated by one element. Then for the inductive step, I supposed that G was finitely generated, say [itex]G = \mathbb{Z} a_{1} + ... + \mathbb{Z} a_{n}[/itex], and that the result holds for groups generated by fewer than n elements. I've then let [itex]H \le G[/itex], and considered the quotient group [itex]G/\mathbb{Z}a_{n}[/itex], and then hoped that the correspondence theorem would help me out, but so far I can't seem to make it work.

Am I even attacking this problem correctly?
 
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  • #2
Try to prove the following general result, it might help:

Let H be an abelian group. If there exists a finitely generated subgroup K of H such that H/K is finitely generated, then H is finitely generated.
 
  • #3
micromass said:
Try to prove the following general result, it might help:

Let H be an abelian group. If there exists a finitely generated subgroup K of H such that H/K is finitely generated, then H is finitely generated.

Does this work?

Let [itex]K \le H[/itex] be finitely generated where [itex]H/K[/itex] is finitely generated, say [itex]K = \mathbb{Z}a_{1} + ... + \mathbb{Z}a_{n}[/itex] and [itex]H/K = \mathbb{Z}(b_{1} + K) + ... + \mathbb{Z}(b_{m} + K) = \mathbb{Z}b_{1} + ... + \mathbb{Z}b_{m} + K[/itex], where the [itex]b_{i} \in H[/itex]. Let [itex]h \in H[/itex], and consider [itex]h + K \in H/K[/itex]. Then [itex]h + K = y_{1}b_{1} + ... + y_{m}b_{m}[/itex] for some [itex]y_{i} \in \mathbb{Z}[/itex], so [itex]y_{1}b_{1} + ... + y_{m}b_{m} - h \in K[/itex]. But then [itex]y_{1}b_{1} + ... + y_{m}b_{m} - h = z_{1}a_{1} + ... + z_{n}a_{n}[/itex] for some [itex]z_{i} \in \mathbb{Z}[/itex], and so [itex]h = y_{1}b_{1} + ... + y_{m}b_{m} - z_{1}a_{1} - ... - z_{n}a_{n}[/itex]. Thus, [itex]H[/itex] is finitely generated.
 
  • #5
Sorry, I'm going to have to ask for a hint as to where to go from there. My ideas for proofs tend to break down when I want to assume [itex]a_{i} \in H[/itex] for some [itex]a_{i}[/itex]when [itex]G = \mathbb{Z}a_{1} + ... + \mathbb{Z}a_{n}[/itex].
 

1. What is a subgroup of a finitely generated abelian group?

A subgroup of a finitely generated abelian group is a subset of elements that form a group, where the group operation and inverse operation are inherited from the larger abelian group. This subgroup is also finitely generated, meaning it can be generated by a finite number of elements.

2. How do you determine if a subgroup is finitely generated?

A subgroup is finitely generated if it can be generated by a finite number of elements. This means that every element in the subgroup can be expressed as a combination of the generators using the group operation and inverse operation.

3. Can a subgroup of a finitely generated abelian group be infinite?

Yes, a subgroup of a finitely generated abelian group can be infinite. As long as the subgroup is generated by a finite number of elements and follows the group properties, it can be infinite in size.

4. What is the significance of a finitely generated abelian group?

Finitely generated abelian groups are important in mathematics and science because they have many useful properties and can be applied to various areas of study, such as algebra, topology, and number theory. They also have connections to other mathematical concepts, such as vector spaces and modules.

5. How are subgroups of finitely generated abelian groups related to cyclic groups?

Subgroups of finitely generated abelian groups can be either cyclic or non-cyclic. If a subgroup is generated by a single element, it is cyclic. However, if it is generated by multiple elements, it is non-cyclic. This is because a cyclic group is a special case of a finitely generated abelian group, where all elements are generated by a single element.

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