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 and $n\ge 3$. Show that the restriction of $\phi$ to $SL(n,F)$ is an outer automorphism of $SL(n,F)$.

  • $\begingroup$ This is related to this question and the linked duplicates. See in particular this question. $\endgroup$ Sep 7, 2017 at 15:13
  • $\begingroup$ For the first link there is no complete and clear answer and for the second one i didn't understand the relation. Please help me. Thanks $\endgroup$
    – aymen
    Sep 7, 2017 at 15:17
  • $\begingroup$ Suppose it were inner, i.e., of the form $hgh^{-1}$ for some $h$. Now compare this with $(g^ {-1})^T$ and choose different elements $g\in SL_n(F)$ and see what you get. Also, first take $n=3$ and compute everything directly. $\endgroup$ Sep 7, 2017 at 15:30
  • $\begingroup$ i think i will fall in this subject i didn't undesrtand what you mean can you please detail more $\endgroup$
    – aymen
    Sep 7, 2017 at 15:38
  • $\begingroup$ Write down some matrices $g$ in $SL(3,F)$ and compute everything! $\endgroup$ Sep 7, 2017 at 16:03


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