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Inner automorphism proof question

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Let G = GL(2, R). Define P : G to G by P(A) = The inverse of the transpose of A (ie. ((A)tr)^-1). P is an automorphism of G, prove that P is not an inner automorphism of G. That is, prove that there does not exist a fixed matrix B in G such that P(A) = BAB^-1 for all A in G. (Hint: consider the restriction of P to the centre Z(G) of G)

    2. Relevant equations
    The centre of G = {A in G: AB = BA for all B in G}

    3. The attempt at a solution
    If we restrict A to Z(G) then we have that BAB^-1 = A for all B in G. So, P(A) = ((A)tr)^-1 = BAB^-1 = A which would imply that the transpose of A equals the inverse of A for all B in G. I'm not sure if this is what we were meant to do, since I can't see how this might help prove P is not an inner autom'sm.

    Any advice would be appreciated.

    W. =)
  2. jcsd
  3. Mar 8, 2010 #2


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    Science Advisor
    Homework Helper

    No, an inner automorphism is defined by P(A)=BAB^(-1) for a fixed B. What you actually have there is P(A)=(transpose(A))^(-1)=A for all A in Z(G). At this point it might be helpful to know what the center of GL(n) is. Or at least some examples of matrices in the center.
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