Unique abelian group of order n

In summary, the conversation discusses determining all integers for which there exists a unique abelian group of a given order. It is suggested that this holds true for prime numbers and a claim is made that for square-free integers, the group \mathbb{Z}_n is the unique abelian group of order n. The conversation then delves into extending this concept to an arbitrary number of factors and discussing the implications of coprime factors.
  • #1
hmw
2
0

Homework Statement



Determine all integers for which there exists a unique abelian group of order n.

Homework Equations





The Attempt at a Solution



All prime integers?
 
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  • #2
Shouldn't you prove it?
 
  • #3
Since this was the first hit when I Googled my homework, I'll resurrect this thread with my thoughts.
This holds trivially for prime numbers.
Claim: For n "square-free", [tex]\mathbb{Z}_n[/tex] is (up to isomorphism) the unique abelian group of order n.

We extend the fact that [tex] \mathbb{Z}_{ab} \cong \mathbb{Z}_a \times \mathbb{Z}_b \Leftrightarrow gcd(a, b) = 1[/tex] by induction to an arbitrary amount of factors. Then the factors (which are also the subscripts of Z, and the orders of the groups) are pairwise coprime, and we have our result.

Thoughts?
 

1. What is a unique abelian group of order n?

A unique abelian group of order n is a mathematical structure consisting of a set of n elements and a binary operation that follows the axioms of an abelian group. This means that the operation is commutative, associative, has an identity element, and every element has an inverse. The term "unique" means that there is only one possible group of this order up to isomorphism.

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

The order of a unique abelian group is simply the number of elements in the group. This can be determined by counting the number of distinct elements or by using the Lagrange's theorem, which states that the order of a subgroup must divide the order of the group.

3. Can a unique abelian group have non-unique subgroups?

Yes, a unique abelian group can have non-unique subgroups. This means that there can be multiple subgroups of the same order within the unique abelian group. However, these subgroups will still be isomorphic to each other and to the unique abelian group itself.

4. How are unique abelian groups related to cyclic groups?

Unique abelian groups are a special type of cyclic group. This means that every unique abelian group is also a cyclic group, but the reverse is not necessarily true. Cyclic groups have a single generator, while unique abelian groups can have multiple generators.

5. Are there any real-world applications of unique abelian groups of order n?

Unique abelian groups of order n have applications in many areas of mathematics, including number theory, graph theory, and cryptography. They can also be used to study symmetry and patterns in nature, such as in crystallography and molecular structures. In addition, unique abelian groups have practical applications in coding theory, error-correcting codes, and signal processing.

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