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Math Help - Orders, Matrices, Complex entries...

  1. #1
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    Orders, Matrices, Complex entries...

    I'm having a little trouble with the difference between my notes and my textbook notation.

    I have

    a = \[ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\]

    b = \[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\]

    I have to determine the orders of \left< a \right> and \left< b \right>, and whether \left< a \right> and \left< b \right> are isomorphic.

    For the first part I have a^4=b^4=I_2

    So they have order 4. Correct?

    For the isomorpic part can I just find a 2x2 matrix that shows that a \rightarrow b isn't a homomorphism?

    Do I even know what I'm talking about? (just started group theory last week)
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  2. #2
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    Quote Originally Posted by MichaelMath View Post
    I'm having a little trouble with the difference between my notes and my textbook notation.

    I have

    a = \[ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\]

    b = \[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\]

    I have to determine the orders of \left< a \right> and \left< b \right>, and whether \left< a \right> and \left< b \right> are isomorphic.

    For the first part I have a^4=b^4=I_2

    So they have order 4. Correct?


    Yes, but also because 4 is the minimal natural power to which both matrices are I

    For the isomorpic part can I just find a 2x2 matrix that shows that a \rightarrow b isn't a homomorphism?

    Do I even know what I'm talking about? (just started group theory last week)
    I'm afraid that the second part makes no much sense: <a>\cong <b> because, as you proved, both these goups are cyclic of the same order (4).

    Tonio
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  3. #3
    MHF Contributor

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    Quote Originally Posted by MichaelMath View Post
    I'm having a little trouble with the difference between my notes and my textbook notation.

    I have

    a = \[ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\]

    b = \[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\]

    I have to determine the orders of \left< a \right> and \left< b \right>, and whether \left< a \right> and \left< b \right> are isomorphic.

    For the first part I have a^4=b^4=I_2

    So they have order 4. Correct?

    For the isomorpic part can I just find a 2x2 matrix that shows that a \rightarrow b isn't a homomorphism?

    Do I even know what I'm talking about? (just started group theory last week)
    If, by " a\rightarrow b" you mean the function that maps a into b, maps a^2 into b^2, a^3 into b^3, and maps a^4= I into b^4= I, that certainly is a homomorphism and, in fact, is the isomorphism you want.

    I don't know what you mean by "find a 2X2 matrix that shows..."
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