Show that the abelian groups are isomorphic

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SUMMARY

The discussion centers on demonstrating the isomorphism between two Abelian groups, P and Q, represented by a 3x3 integer matrix A and its transpose B. The key approach involves reducing both matrices to diagonal form and analyzing the effects of row and column operations on A and B. The participants emphasize the necessity of precise definitions regarding the representation of Abelian groups in this context to facilitate a clearer understanding of the problem.

PREREQUISITES
  • Understanding of Abelian groups and their properties
  • Familiarity with matrix operations, specifically row and column operations
  • Knowledge of matrix transposition and its implications
  • Experience with diagonalization of matrices
NEXT STEPS
  • Study the properties of Abelian groups in detail
  • Learn about matrix diagonalization techniques
  • Explore the effects of row and column operations on matrix transformations
  • Investigate the concept of isomorphism in group theory
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those studying abstract algebra, as well as students and educators seeking to deepen their understanding of group theory and matrix operations.

buckylomax
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Hi there,

I'm trying to figure out this question:

Let A=[aij] be a 3x3 matrix with integer entries and let B=[bij] be it’s transpose. Let P and Q be the Abelian groups represented by A and B respectively. Show that P and Q are isomorphic by comparing the effects of row and column operations on A and B.

I've very stuck with this question. I figure I need to reduce both matrices to diagonal form and then compare them but I'm not sure how to get there. Any advices would be appreciated.

Thanks

B.
 
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Can you be more precise about what you mean by "the Abelian groups represented by A and B"?
 

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