# Element structure of general linear group:GL(2,3)

## Contents

This article gives specific information, namely, element structure, about a particular group, namely: general linear group:GL(2,3).
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This article describes the element structure of general linear group:GL(2,3) which is the general linear group of degree two over field:F3.

## Conjugacy class structure

### Interpretation as general linear group of degree two over field:F3

Nature of conjugacy class Eigenvalues Characteristic polynomial Minimal polynomial Size of conjugacy class (generic $q$) Size of conjugacy class ($q = 3$) Number of such conjugacy classes (generic $q$) Number of such conjugacy classes ($q = 3$) Total number of elements (generic $q$) Total number of elements ($q = 3$) Representative matrix (one per conjugacy class)
Diagonalizable over $\mathbb{F}_3$ with equal diagonal entries, hence a scalar $\{a,a \}$ where $a \in \mathbb{F}_3^\ast$ $(x - a)^2$ where $a \in \mathbb{F}_3^\ast$ $x - a$ where $a \in \mathbb{F}_3^\ast$ 1 1 $q - 1$ 2 $q - 1$ 2 $\begin{pmatrix}1 & 0 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix}-1 & 0 \\ 0 & -1 \\\end{pmatrix}$
Diagonalizable over field:F9, not over $\mathbb{F}_3$. Must necessarily have no repeated eigenvalues. Pair of conjugate elements of $\mathbb{F}_{9}$ $x^2 + 1$, $x^2 + x - 1$, $x^2 - x - 1$ Same as characteristic polynomial $q(q - 1)$ 6 $q(q-1)/2$ 3 $q^2(q-1)^2/2$ 18 $\begin{pmatrix} 0 & 1 \\ -1 & 0 \\\end{pmatrix}$, $\begin{pmatrix} 0 & 1 \\ 1 & -1 \\\end{pmatrix}$, $\begin{pmatrix} 0 & 1 \\ 1 & 1 \\\end{pmatrix}$
Not diagonal, has Jordan block of size two $a$ (multiplicity two) where $a \in \mathbb{F}_3^\ast$ $(x - a)^2$ where $a \in \mathbb{F}_3^\ast$ Same as characteristic polynomial $q^2 - 1$ 8 $q - 1$ 2 $(q + 1)(q - 1)^2$ 16 $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}$
Diagonalizable over field:F3 with distinct diagonal entries $\{ 1, -1 \}$ $x^2 - 1$ Same as characteristic polynomial $q(q + 1)$ 12 $(q - 1)(q - 2)/2$ 1 $q(q+1)(q-1)(q-2)/2$ 12 $\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$
Total NA NA NA NA NA $q^2 - 1$ 8 $q^4 - q^3 - q^2 + q$ 48 NA