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Math Help - Isomorphsim

  1. #1
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    Isomorphsim

    I would like to know if O(n) isomorphic to the product group SO(n) ×{ħI}?
    where O(n) is orthogonal group
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  2. #2
    Super Member girdav's Avatar
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    Re: Isomorphsim

    O(n) has two connected components, namely, the set of matrices of determinant 1 and those of determinant -1. So you can define f(A)=(A,I) if \det A and f(A)=(-A,-I) otherwise.
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  3. #3
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    Re: Isomorphsim

    yes. SO(n) is normal in O(n), being of index 2 (because det:O(n)-->{-1,1} is a surjective group homomorphism).

    it should be clear that we thus get two cosets: SO(n), and -SO(n) (since if a matrix A = BU is not in SO(n), where U is in SO(n), then:

    det(A) = det(B)det(U) = det(B) = -1, so we have det(-A) = det(-IA) = det(-I)det(A) = (-1)(-1) = 1, so -A is in SO(n), and A = -(-A)).

    this shows we can write any matrix in O(n) UNIQUELY as a product of a matrix in SO(n) and either I or -I.

    therefore O(n) is the (internal) direct product of SO(n) and {-I,I} (the above shows it is the semi-direct product of SO(n) and {-I,I}. but let A be an orthogonal nxn matrix. then:

    IAI-1 = IAI = A, and:

    (-I)A(-I)-1 = IAI = A,

    which shows that {-I,I} acts trivially on O(n) by conjugation, so we have a direct product. alternatively, you can show that {-I,I} is normal in O(n) by direct computation:

    AIA-1 = AA-1 = I, and A(-I)A-1 = -AA-1 = -I).
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