Answer to Question #122425 in Abstract Algebra for Saleem

Question

Compute the center of GL(3.F) Also find order of GL(3,3)

Answer 1

$GL(3,F)=\\=\left\{A=\begin{pmatrix} a_{11} & a_{12}&a_{13} \\ a_{21} & a_{22}&a_{23} \\ a_{31} & a_{32}&a_{33} \\ \end{pmatrix}, a_{ij}\in F, detA=1\right\}$

The center is

$Z=\left\{z\in GL(3,F):\forall A\in GL(3,F):zA=Az\right\}\\ z=\begin{pmatrix} b_{11} & b_{12}&b_{13} \\ b_{21} & b_{22}&b_{23} \\ b_{31} & b_{32}&b_{33} \\ \end{pmatrix}\\ zA=\begin{pmatrix} b_{11} & b_{12}&b_{13} \\ b_{21} & b_{22}&b_{23} \\ b_{31} & b_{32}&b_{33} \\ \end{pmatrix}\cdot\begin{pmatrix} a_{11} & a_{12}&a_{13} \\ a_{21} & a_{22}&a_{23} \\ a_{31} & a_{32}&a_{33} \\ \end{pmatrix}\\ Az=\begin{pmatrix} a_{11} & a_{12}&a_{13} \\ a_{21} & a_{22}&a_{23} \\ a_{31} & a_{32}&a_{33} \\ \end{pmatrix}\cdot \begin{pmatrix} b_{11} & b_{12}&b_{13} \\ b_{21} & b_{22}&b_{23} \\ b_{31} & b_{32}&b_{33} \\ \end{pmatrix}$

We get the system

$a_{21}b_{12}+a_{31}b_{13}=a_{12}b_{21}+a_{13}b_{31}\\ a_{11}b_{21}+a_{21}b_{22}+a_{31}b_{23}=a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31}\\ a_{11}b_{31}+a_{21}b_{32}+a_{31}b_{33}=a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31}\\ a_{12}b_{11}+a_{22}b_{12}+a_{32}b_{13}=a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}\\ a_{12}b_{21}+a_{32}b_{23}=a_{21}b_{12}+a_{23}b_{32}\\ a_{12}b_{31}+a_{22}b_{32}+a_{32}b_{33}=a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32}\\ a_{13}b_{11}+a_{23}b_{12}+a_{33}b_{13}=a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33}\\ a_{13}b_{21}+a_{23}b_{22}+a_{33}b_{23}=a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\ a_{13}b_{31}+a_{23}b_{32}=a_{31}b_{13}+a_{32}b_{23}$

Since $detA\neq0$ , the system has solutions if

$a_{11}\neq0, a_{22}\neq0, a_{33}\neq0, a_{ij}=0, i\neq j$

$z=\begin{pmatrix} a_{11} &0&0 \\ 0 &a_{22}&0\\ 0&0&a_{33} \end{pmatrix}$

Order $GL(3,Z_3)$

$(p^n-p^{n-1})(p^n-p^{n-2})(p^n-p^{n-3})...=\\ =(3^3-3^2)(3^3-3)(3^3-1)=18\cdot24\cdot26=11232$