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Math Help - Problem on inner automorphism

  1. #1
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    Problem on inner automorphism

    f: GLn(R)-->GLn(R) is defined by f(A)= (A-1)T . Show that f is not an
    inner automorphism.

    My Solution:
    Assume that there is C in GLn(R) such that CAC-1 = (A-1)T for all A in GLn(R).
    Then for all A in GLn(R)
    det(CAC-1) = det (A-1)T
    => det(C). det(A). Det(C-1) = det(A-1)
    => det(A) = det(A-1).

    But, the above is not true for all A in GLn(R).
    Therefore f is not an inner automorphism of GLn(R).

    Is this correct?
    Please help!
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Problem on inner automorphism

    Yes, your proof is correct.

    You could also provide a counterexample. For instance, let n=2 and A=\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}. Then \left(A^{-1}\right)^{\mathrm T}=\begin{pmatrix} \frac12 & 0 \\ 0 & \frac12 \end{pmatrix}. Thus \det CAC^{-1}=\det A=4 for any C\in\mathrm{GL}_2(\mathbb R) whereas \det\left(A^{-1}\right)^{\mathrm T}=\frac14\ne\det A.
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  3. #3
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    Re: Problem on inner automorphism

    Quote Originally Posted by Nehushtan View Post
    Yes, your proof is correct.

    You could also provide a counterexample. For instance, let n=2 and A=\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}. Then \left(A^{-1}\right)^{\mathrm T}=\begin{pmatrix} \frac12 & 0 \\ 0 & \frac12 \end{pmatrix}. Thus \det CAC^{-1}=\det A=4 for any C\in\mathrm{GL}_2(\mathbb R) whereas \det\left(A^{-1}\right)^{\mathrm T}=\frac14\ne\det A.
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