Show that the abelian groups are isomorphic

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In summary, the conversation discusses the comparison of two matrices A and B and how the effects of row and column operations on these matrices can show that the Abelian groups represented by them (P and Q, respectively) are isomorphic. The person asking the question is struggling with reducing the matrices to diagonal form and asks for advice.
  • #1
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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  • #2
Can you be more precise about what you mean by "the Abelian groups represented by A and B"?
 

FAQ: Show that the abelian groups are isomorphic

What is an abelian group?

An abelian group is a mathematical structure that satisfies the commutative property, meaning that the order in which the group's operations are performed does not affect the outcome. In simpler terms, the result of adding or multiplying elements in an abelian group is the same regardless of their order.

What does it mean for two groups to be isomorphic?

Two groups are said to be isomorphic if there exists a bijective function between them that preserves their group structure. This means that the two groups have the same number of elements and the same operations, and the only difference between them is the way their elements are labeled or represented.

How do you prove that two abelian groups are isomorphic?

To prove that two abelian groups are isomorphic, you need to show that there exists a bijective function between them that preserves the group structure. This can be done by finding a mapping between the elements of the two groups that satisfies the properties of a homomorphism, which is a function that preserves the group's operation. You also need to show that the function is one-to-one and onto, meaning that each element in one group is mapped to a unique element in the other group.

What are some common examples of abelian groups?

Some common examples of abelian groups include the integers (under addition), rational numbers (under addition), real numbers (under addition), and complex numbers (under addition). Other examples include the group of invertible matrices, the group of symmetries of a regular polygon, and the group of rotations in three-dimensional space.

Why are abelian groups important in mathematics?

Abelian groups play a crucial role in mathematics because they provide a fundamental framework for understanding and classifying different mathematical structures. They have many important applications in various areas of mathematics, including algebra, geometry, number theory, and topology. Additionally, the study of abelian groups has led to many important discoveries and developments in modern mathematics.

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