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I need help to answer the following problem:

Let $F$ be a field and $n\ge 2$.

Define $\phi:GL(n,F)\to GL(n,F)$ by $\phi(g)=(g^{-1})^T$, where $T$ denotes the transpose.

Suppose that $F$ has at least four elements. Show that $\phi$ is an outer automorphism of $GL(n,F)$.

Thanks in advance for your help.

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    Hint: Consider the center of the group. What happens to the center under an inner automorphism? – Tobias Kildetoft Sep 06 '17 at 18:46
  • i didn't understand the answer can you please explain more. Thanks –  Sep 06 '17 at 19:23
  • Which part of the answer at the duplicate you do not understand? That any element of the center must be fixed, for an inner automorphism? That $\phi$ above does not fix it? – Dietrich Burde Sep 06 '17 at 19:34
  • That $ϕ$ above does not fix it? –  Sep 06 '17 at 19:49
  • Mona, do you know what the center of $GL(n,F)$ is? – Jyrki Lahtonen Sep 06 '17 at 20:36
  • the centre of $\operatorname{GL}(n,F)$ is $\lbrace\lambda I\mid λ ∈ F^∗\rbrace$. and then what to do? –  Sep 06 '17 at 20:52
  • Ok. What is $((\lambda I)^{-1})^T$? Compare with $A(\lambda I)A^{-1}$ for some arbitrary $A\in GL(n,F)$. – Jyrki Lahtonen Sep 07 '17 at 07:40
  • $((\lambda I)^{-1})^T=\frac{1}{\lambda}I$ and $A(\lambda I)A^{-1} =\lambda I$ then $((\lambda I)^{-1})^T$ is different from $A(\lambda I)A^{-1}$ for all $\lambda\in F\backslash\left{1_F,-1_F\right}$. and how does that answer the quetion? –  Sep 07 '17 at 14:28
  • Mona, if $F$ has at least four elements, then there exists $\lambda\in F\setminus{\pm1_F,0_F}$. – Jyrki Lahtonen Sep 08 '17 at 19:04

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