Are G & H Isomorphic? Prove Answer | Group of 2x2 Matrices

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In summary, the conversation is discussing whether the groups G and H, consisting of 2x2 invertible upper and lower triangular matrices respectively, are isomorphic. The idea of using transposition as an isomorphism is suggested, but it is noted that transposition is not a homomorphism. The possibility of combining transposition with another operation is considered. Ultimately, it is concluded that G and H are not isomorphic and the reasoning behind this is explained.
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Let G be the group of 2x2 invertible upper triangular matrices and H be the group of 2x2 invertible lower triangular matrices (both groups under multiplication). Are G and H isomorphic? Prove answer.

First I thought they were isomorphic, but couldn't find an isomorphism, so now I believe they are not isomorphic, but can't pinpoint exactly why.
Any suggestions?
 
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Transposition is a good first guess for an isomorphism, since the transpose of an upper triangular matrix is lower triangular, and vice-versa. But transposition isn't a homomorphism, since [tex](AB)^T=B^TA^T[/tex] (the order gets reversed). Kind of like how [tex](AB)^{-1}=B^{-1}A^{-1}[/tex]. Can you combine them?
 
  • #3
Tinyboss, Why do you say it is a good first guess. As you mention, the order gets reversed.
I don't think they are isomorphic. I know that if there was an isomorphism f, then f(-A)= -f(A) for all matrices A in G. Also, f would map the subgroup D of diagonal matrices to itself.
 
  • #4
Read the last sentence in tinyboss's reply...
 
  • #5
Ohh, of course.
Thank you guys.
 

1. What is an isomorphism?

An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical objects. In the context of group theory, it refers to a mapping between two groups that preserves the group structure and operations.

2. How do you determine if two groups are isomorphic?

To determine if two groups are isomorphic, you must find a bijective mapping between the groups that preserves the group structure and operations. In the case of 2x2 matrices, this would involve finding a bijective mapping between the two groups that preserves the matrix multiplication and addition operations.

3. What is the group of 2x2 matrices?

The group of 2x2 matrices, denoted as GL(2), is a mathematical group that consists of all invertible 2x2 matrices. It is also referred to as the general linear group of order 2. This group has a total of 6 elements.

4. How do you prove that two groups of 2x2 matrices are isomorphic?

To prove that two groups of 2x2 matrices are isomorphic, you must show that there exists a bijective mapping between the two groups that preserves the group structure and operations. This can be done by explicitly defining the mapping and showing that it satisfies the necessary conditions.

5. What is the significance of proving that two groups of 2x2 matrices are isomorphic?

Proving that two groups of 2x2 matrices are isomorphic is significant in group theory as it allows us to understand the underlying structure and properties of these groups. It also allows us to apply theorems and results from one group to the other, making problem solving and calculations more efficient.

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