MHB Show that the abelian groups are isomorphic

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To show that the Abelian groups P and Q, represented by the 3x3 integer matrices A and its transpose B, are isomorphic, one must analyze the effects of row and column operations on both matrices. The key is to reduce both A and B to diagonal form, which allows for a direct comparison of their structure. Clarification is needed on what is meant by "the Abelian groups represented by A and B" to ensure a proper understanding of the groups in question. Understanding the relationship between the row operations on A and the column operations on B is crucial for establishing the isomorphism. The discussion emphasizes the importance of precise definitions and methods in proving the isomorphism of these groups.
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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