Unique abelian group of order n

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SUMMARY

The unique abelian group of order n exists if and only if n is square-free or a prime number. For prime integers, the unique abelian group is simply the cyclic group \(\mathbb{Z}_p\). The claim established in the discussion states that for square-free integers, \(\mathbb{Z}_n\) is the unique abelian group up to isomorphism. This conclusion is derived from the property that \(\mathbb{Z}_{ab} \cong \mathbb{Z}_a \times \mathbb{Z}_b\) if and only if gcd(a, b) = 1, which can be extended inductively to any number of coprime factors.

PREREQUISITES
  • Understanding of abelian groups and their properties
  • Familiarity with the concept of square-free integers
  • Knowledge of group isomorphism and cyclic groups
  • Basic number theory, particularly gcd and prime factorization
NEXT STEPS
  • Study the structure of abelian groups in detail
  • Learn about the classification of finite abelian groups
  • Explore the implications of the Fundamental Theorem of Finite Abelian Groups
  • Investigate examples of square-free integers and their corresponding abelian groups
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, group theory enthusiasts, and educators looking to deepen their understanding of abelian group structures.

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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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Shouldn't you prove it?
 
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?
 

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