Is H a Free Commutative Group of Rank n in Z^n?

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SUMMARY

The discussion centers on the characterization of subgroup H in Z^n, establishing that H is a subgroup of finite index in Z^n if and only if H is a free commutative group of rank n. Participants reference the elementary divisors theorem as a foundational concept for understanding finite generation in abelian groups. The conversation emphasizes the importance of categorizing this topic within the realm of Abstract Algebra rather than science education.

PREREQUISITES
  • Understanding of free commutative groups
  • Familiarity with subgroup index concepts
  • Knowledge of Z^n and its properties
  • Elementary divisors theorem for finite generation in abelian groups
NEXT STEPS
  • Study the properties of free commutative groups in Abstract Algebra
  • Explore subgroup index calculations in Z^n
  • Review the elementary divisors theorem in detail
  • Investigate the implications of finite index subgroups in group theory
USEFUL FOR

Mathematicians, particularly those specializing in Abstract Algebra, students studying group theory, and anyone interested in the properties of finite index subgroups in abelian groups.

charlamov
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show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n
 
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charlamov said:
show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n


What've you done? Do you know the elementary divisors theorem for fin. gen, abelian groups?

DonAntonio
 
charlamov said:
show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n



THis question doesn't belong at all in science education, but in "Abstract Algebra" in the math department
 

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