G L ( 3 , F ) = = { A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , a i j ∈ F , d e t A = 1 } GL(3,F)=\\=\left\{A=\begin{pmatrix}
a_{11} & a_{12}&a_{13} \\
a_{21} & a_{22}&a_{23} \\
a_{31} & a_{32}&a_{33} \\
\end{pmatrix},
a_{ij}\in F, detA=1\right\} G L ( 3 , F ) = = ⎩ ⎨ ⎧ A = ⎝ ⎛ a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 ⎠ ⎞ , a ij ∈ F , d e t A = 1 ⎭ ⎬ ⎫
The center is
Z = { z ∈ G L ( 3 , F ) : ∀ A ∈ G L ( 3 , F ) : z A = A z } z = ( b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ) z A = ( b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ) ⋅ ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) A z = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) ⋅ ( b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ) Z=\left\{z\in GL(3,F):\forall A\in GL(3,F):zA=Az\right\}\\
z=\begin{pmatrix}
b_{11} & b_{12}&b_{13} \\
b_{21} & b_{22}&b_{23} \\
b_{31} & b_{32}&b_{33} \\
\end{pmatrix}\\
zA=\begin{pmatrix}
b_{11} & b_{12}&b_{13} \\
b_{21} & b_{22}&b_{23} \\
b_{31} & b_{32}&b_{33} \\
\end{pmatrix}\cdot\begin{pmatrix}
a_{11} & a_{12}&a_{13} \\
a_{21} & a_{22}&a_{23} \\
a_{31} & a_{32}&a_{33} \\
\end{pmatrix}\\
Az=\begin{pmatrix}
a_{11} & a_{12}&a_{13} \\
a_{21} & a_{22}&a_{23} \\
a_{31} & a_{32}&a_{33} \\
\end{pmatrix}\cdot \begin{pmatrix}
b_{11} & b_{12}&b_{13} \\
b_{21} & b_{22}&b_{23} \\
b_{31} & b_{32}&b_{33} \\
\end{pmatrix} Z = { z ∈ G L ( 3 , F ) : ∀ A ∈ G L ( 3 , F ) : z A = A z } z = ⎝ ⎛ b 11 b 21 b 31 b 12 b 22 b 32 b 13 b 23 b 33 ⎠ ⎞ z A = ⎝ ⎛ b 11 b 21 b 31 b 12 b 22 b 32 b 13 b 23 b 33 ⎠ ⎞ ⋅ ⎝ ⎛ a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 ⎠ ⎞ A z = ⎝ ⎛ a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 ⎠ ⎞ ⋅ ⎝ ⎛ b 11 b 21 b 31 b 12 b 22 b 32 b 13 b 23 b 33 ⎠ ⎞
We get the system
a 21 b 12 + a 31 b 13 = a 12 b 21 + a 13 b 31 a 11 b 21 + a 21 b 22 + a 31 b 23 = a 21 b 11 + a 22 b 21 + a 23 b 31 a 11 b 31 + a 21 b 32 + a 31 b 33 = a 31 b 11 + a 32 b 21 + a 33 b 31 a 12 b 11 + a 22 b 12 + a 32 b 13 = a 11 b 12 + a 12 b 22 + a 13 b 32 a 12 b 21 + a 32 b 23 = a 21 b 12 + a 23 b 32 a 12 b 31 + a 22 b 32 + a 32 b 33 = a 31 b 12 + a 32 b 22 + a 33 b 32 a 13 b 11 + a 23 b 12 + a 33 b 13 = a 11 b 13 + a 12 b 23 + a 13 b 33 a 13 b 21 + a 23 b 22 + a 33 b 23 = a 21 b 13 + a 22 b 23 + a 23 b 33 a 13 b 31 + a 23 b 32 = a 31 b 13 + a 32 b 23 a_{21}b_{12}+a_{31}b_{13}=a_{12}b_{21}+a_{13}b_{31}\\
a_{11}b_{21}+a_{21}b_{22}+a_{31}b_{23}=a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31}\\
a_{11}b_{31}+a_{21}b_{32}+a_{31}b_{33}=a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31}\\
a_{12}b_{11}+a_{22}b_{12}+a_{32}b_{13}=a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}\\
a_{12}b_{21}+a_{32}b_{23}=a_{21}b_{12}+a_{23}b_{32}\\
a_{12}b_{31}+a_{22}b_{32}+a_{32}b_{33}=a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32}\\
a_{13}b_{11}+a_{23}b_{12}+a_{33}b_{13}=a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33}\\
a_{13}b_{21}+a_{23}b_{22}+a_{33}b_{23}=a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\
a_{13}b_{31}+a_{23}b_{32}=a_{31}b_{13}+a_{32}b_{23} a 21 b 12 + a 31 b 13 = a 12 b 21 + a 13 b 31 a 11 b 21 + a 21 b 22 + a 31 b 23 = a 21 b 11 + a 22 b 21 + a 23 b 31 a 11 b 31 + a 21 b 32 + a 31 b 33 = a 31 b 11 + a 32 b 21 + a 33 b 31 a 12 b 11 + a 22 b 12 + a 32 b 13 = a 11 b 12 + a 12 b 22 + a 13 b 32 a 12 b 21 + a 32 b 23 = a 21 b 12 + a 23 b 32 a 12 b 31 + a 22 b 32 + a 32 b 33 = a 31 b 12 + a 32 b 22 + a 33 b 32 a 13 b 11 + a 23 b 12 + a 33 b 13 = a 11 b 13 + a 12 b 23 + a 13 b 33 a 13 b 21 + a 23 b 22 + a 33 b 23 = a 21 b 13 + a 22 b 23 + a 23 b 33 a 13 b 31 + a 23 b 32 = a 31 b 13 + a 32 b 23
Since d e t A ≠ 0 detA\neq0 d e t A = 0 , the system has solutions if
a 11 ≠ 0 , a 22 ≠ 0 , a 33 ≠ 0 , a i j = 0 , i ≠ j a_{11}\neq0, a_{22}\neq0, a_{33}\neq0, a_{ij}=0, i\neq j a 11 = 0 , a 22 = 0 , a 33 = 0 , a ij = 0 , i = j
z = ( a 11 0 0 0 a 22 0 0 0 a 33 ) z=\begin{pmatrix}
a_{11} &0&0 \\
0 &a_{22}&0\\
0&0&a_{33}
\end{pmatrix} z = ⎝ ⎛ a 11 0 0 0 a 22 0 0 0 a 33 ⎠ ⎞
Order G L ( 3 , Z 3 ) GL(3,Z_3) G L ( 3 , Z 3 )
( p n − p n − 1 ) ( p n − p n − 2 ) ( p n − p n − 3 ) . . . = = ( 3 3 − 3 2 ) ( 3 3 − 3 ) ( 3 3 − 1 ) = 18 ⋅ 24 ⋅ 26 = 11232 (p^n-p^{n-1})(p^n-p^{n-2})(p^n-p^{n-3})...=\\
=(3^3-3^2)(3^3-3)(3^3-1)=18\cdot24\cdot26=11232 ( p n − p n − 1 ) ( p n − p n − 2 ) ( p n − p n − 3 ) ... = = ( 3 3 − 3 2 ) ( 3 3 − 3 ) ( 3 3 − 1 ) = 18 ⋅ 24 ⋅ 26 = 11232
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