Proving H is Cyclic: Finite Abelian Group

In summary, we want to prove that a finite abelian group H, with a subgroup of order d for every positive divisor d of the order of H, is cyclic. To do this, we can assume the opposite and use the Fundamental theorem of finitely generated abelian groups to show that H is isomorphic to Z_a x Z_b, where a and b are not relatively prime. This leads us to consider applying Cauchy's theorem.
  • #1

Homework Statement


Let H be a finite abelian group that has one subgroup of order d for every positive divisor d of the order of H. Prove that H is cyclic

Homework Equations


We want to show H={a^n|n is an integer}
 
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  • #2
I haven't done this stuff in a while but since no one is helping I'll give it a shot.

Assume not. Then by Fundamental theorem of finitely generated abelian groups, H is isomorphic to Z_a x Z_b. Since, by assumption, H is not cyclic, it follows that Z_a x Z_b is not cyclic. This implies that a and b are not relatively prime. Which implies that there exists some prime p that divides both a and b. Perhaps apply Cauchy's theorem here.
 

1. What is a cyclic group?

A cyclic group is a mathematical structure that consists of a set of elements and an operation that combines any two elements to create a third element. In a cyclic group, there is a specific element called the generator that can be used to create all other elements within the group through repeated application of the operation.

2. How can you prove that a group is cyclic?

In order to prove that a group is cyclic, you need to show that there is a specific element within the group that can generate all other elements through repeated application of the group's operation. For finite abelian groups, this can be done by showing that the group has a prime power order and that all its elements have the same order.

3. What is the significance of H being a finite abelian group in this proof?

The proof for showing that H is cyclic relies on the fact that H is a finite abelian group. This means that the group has a finite number of elements and that its operation is commutative, which simplifies the process of finding a generator element and proving that it can generate all other elements in the group.

4. Can you give an example of a finite abelian group that is not cyclic?

Yes, an example of a finite abelian group that is not cyclic is the Klein four-group, which has four elements and an operation that is commutative. However, none of its elements can generate all other elements within the group through repeated application of the operation.

5. How does proving that H is cyclic relate to other properties of groups?

Proving that H is cyclic is an important step in understanding the structure of the group and its properties. It can help determine other properties of the group, such as whether it is a normal subgroup, if it has any subgroups, and the order of its elements. It also allows for the use of specific theorems and techniques that apply to cyclic groups.

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