inner automorphism

Question

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.

Show that the restriction of $\phi$ to $SL(2,F)$ is an inner automorphism of $SL(2,F)$. Thanks in advance

Please do not repost the question again. You could add there $n=2$. For size $2$ a direct computation is really easy. — Dietrich Burde, Sep 07 '17 at 16:04

score 0 · Answer 1 · answered Sep 07 '17 at 16:04

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I do not have enough reputation to comment. Therefore I have to write my comment as an answer.

An inner automorphism $\phi$ of a group $G$ is a map $\phi(g)=aga^{-1}$ with $a,g\in G$. In the case of $G=SL(2,F)$, can you find an $a\in SL(2,F)$ such that $(g^{-1})^{T}=aga^{-1}$by trial and error?

answered Sep 07 '17 at 16:04

Dianbin Bao

177

do you mean that the question is false? – aymen Sep 07 '17 at 16:09
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I did not mean it is false. Have you tried some examples? For instance, what happens if $a=\left(\begin{matrix}0,-1,\1,0\end{matrix}\right)$ – Dianbin Bao Sep 07 '17 at 16:13
for this $a$ we have the equality for all $g\in G$ – aymen Sep 07 '17 at 16:18

score 0 · Answer 2 · answered Sep 07 '17 at 16:28

On $SL(2,F)$, we have that the standard formula for the inverse of a matrix takes a simpler form since the determinant is one. On $SL(2,F)$, $\phi$ is just conjugation by the matrix $$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$ as you can easily check. This shows $\phi$ restricted to $SL(2,F)$ is an inner automorphism.

inner automorphism

2 Answers2