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

aymen
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  • This is related to this question and the linked duplicates. See in particular this question. – Dietrich Burde Sep 07 '17 at 15:13
  • 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 – aymen Sep 07 '17 at 15:17
  • 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. – Dietrich Burde Sep 07 '17 at 15:30
  • i think i will fall in this subject i didn't undesrtand what you mean can you please detail more – aymen Sep 07 '17 at 15:38
  • Write down some matrices $g$ in $SL(3,F)$ and compute everything! – Dietrich Burde Sep 07 '17 at 16:03
  • $g=\begin{pmatrix}g_{11}&g_{12}&g_{13}\g_{21}&g_{22}&g_{23}\g_{31}&g_{32}&g_{33}\end{pmatrix}$ with $det g=1$ – aymen Sep 07 '17 at 16:31
  • let us suppose that it were inner ie the form $hgh^{-1}$. and let us take to ellements $g_1,g_2\in G$ then $hg_1h^{-1}=(g^{-1}_1)^{T}$ and $hg_2h^{-1}=(g^{-1}_2)^{T}$ and then what should i do? – aymen Sep 07 '17 at 16:43
  • Choose specific matrices $g$, with, say, only $0$ and $1$ as coefficients. Assume always that $(g^{-1})^T=hgh^{-1}$ with the same $h$. Then you get finally a contradiction. For computation it is better to write it as $(g^{-1})^Th=hg$, so you do not have to compute an inverse. – Dietrich Burde Sep 07 '17 at 16:46
  • and what about all $n\ge 3$? – aymen Sep 07 '17 at 16:52
  • what are the spesific matrix in the general cas $n\ge 3$? – aymen Sep 07 '17 at 17:28
  • Suppose it were inner, ie there exists $h\in SL(n,F)$ such that $\phi(g)=hgh^{−1}$ for all $g\in SL(n,F)$. we take then $g\in SL_n(F)$ such that $gh=hg$ and $g\neq (g^{-1})^T$ then we get a contradiction since $\phi(g)=hgh^{−1}=g$ is different from $(g^{-1})^T$ – aymen Sep 08 '17 at 14:38
  • Suppose it were inner, ie there exists $h∈SL(n,F)$such that $ϕ(g)=hgh^{−1}$ for all $g∈SL(n,F)$. we take then specific matrices $g_1,g_2 ∈SLn(F)$with only $0$ and $1$ as coefficients then we get a contradiction: $ϕ(g_1)=hg_1h^{−1}=(g_1^{-1})^T$ and $ϕ(g_2)=hg_2h^{−1}=(g_2^{-1})^T$ then $hg_1h^{−1}hg_2h^{−1}=(g_1^{-1})^T(g_2^{-1})^T$ then $hg_1g_2h^{−1}=(g_1^{-1})^T(g_2^{-1})^T$ wich give the contadiction since we dont have the same coefficients in the lift and right side of the final equality – aymen Sep 08 '17 at 18:04

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