<s>[INST] Representation of Quaternion group in $GL(2,3)$ I am working with the representation of the quaternion group in $GL(2,3)$ generated by $A=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}, B=\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}, C=\begin{pmatrix} -1 & 1\\ 1 & 1 \end{pmatrix}$.  
Now $C=PBP^{-1}$ where $P= \begin{pmatrix} 0 & 1\\ 1& 0 \end{pmatrix}$.  I am trying to show that $A$ and $B$ are similar by an element in $GL(2,3)$. 
They are obviously similar in $GL(2,9)$ because they are both diagonalizable with the same eigenvalues:  Their characteristic polynomials are both $x^2+1$, and since its irreducible over $F_3$ and $F_3$ is perfect, the roots are distinct.   
I feel that because of the relations $A=CB, B=AC, C=BA$,    $A$, $B$, $C$ should all "be on the same footing": $B$ and $C$ similar by P should mean that $A,B$ are similar too, without me having to worry about whether the similarity matrix has entries in $F_3$.  After all I should be able to use jacobi's identity and the matrix P to achieve this.  It has not been working out for some time!
edit:
another thing I am trying -
Probably the best way to do this is to look at all the embeddings of $Q_8 \hookrightarrow GL(2,3)$.  
 [/INST] Let's suppose they are conjugate by 
$$M = \left(\begin{array}{rr}
x & y\\
z & w \end{array} \right).$$
Then, we have
$$MB = AM \Rightarrow \left(\begin{array}{rr}
x + y & x - y\\
z+w & z-w \end{array} \right) = \left(\begin{array}{rr}
-z & -w\\
x & y \end{array} \right).$$
So, $z = -x- y, w = y - x$ and putting these into the last two equations gives
$$x = x, y = y$$
and hence we can choose arbitrary values of $x$ and $y$. So define $M$ as
$$M = \left(\begin{array}{rr}
1 & 0\\
-1 & -1 \end{array} \right).$$
Just for confirmation, 
$$MB = \left(\begin{array}{rr}
1 & 1\\
1 & 0 \end{array} \right)$$
and
$$AM =\left(\begin{array}{rr}
1 & 1\\
1 & 0 \end{array} \right).$$
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