# Element structure of projective general linear group:PGL(2,9)

## Contents

This article gives specific information, namely, element structure, about a particular group, namely: projective general linear group:PGL(2,9).
View element structure of particular groups | View other specific information about projective general linear group:PGL(2,9)

This article describes the element structure of projective general linear group:PGL(2,9), which is the projective general linear group of degree two over field:F9.

Note that this group is not isomorphic to symmetric group:S6, even though it has a subgroup of index two that is isomorphic to alternating group:A6.

## Conjugacy class structure

There is a total of $11$ conjugacy classes.

Nature of conjugacy class upstairs in $GL_2$ Eigenvalues Characteristic polynomial Minimal polynomial Size of conjugacy class Number of such conjugacy classes Total number of elements
Diagonalizable over field:F9 with equal diagonal entries, hence a scalar $\{ a, a \}$ where $a \in \mathbb{F}_9^\ast$ $(x - a)^2$ $x - a$ 1 1 1
Diagonalizable over $\mathbb{F}_{81}$, not over $\mathbb{F}_9$, eigenvalues are negatives of each other. Pair of mutually negative conjugate elements of $\mathbb{F}_{81}$. All such pairs identified. $x^2 - \mu$, $\mu$ a nonzero non-square Same as characteristic polynomial 36 1 $q(q - 1)/2$
Diagonalizable over $\mathbb{F}_9$ with mutually negative diagonal entries. $\{ \lambda, - \lambda \}$, all such pairs identified. $x^2 - \lambda^2$, all identified Same as characteristic polynomial 45 1 45
Diagonalizable over $\mathbb{F}_{81}$, not over $\mathbb{F}_9$, eigenvalues are not negatives of each other. Pair of conjugate elements of $\mathbb{F}_{81}$. Each pair identified with anything obtained by multiplying both elements of it by an element of $\mathbb{F}_9$. $x^2 - ax + b$, $a \ne 0$, irreducible; with identification. Same as characteristic polynomial 72 4 288
Not diagonal, has Jordan block of size two $a \in\mathbb{F}_9^\ast$ (multiplicity 2). Each conjugacy class has one representative of each type. $(x - a)^2$ Same as characteristic polynomial 80 1 80
Diagonalizable over $\mathbb{F}_9$ with distinct diagonal entries whose sum is not zero. $\lambda, \mu$ where $\lambda,\mu \in \mathbb{F}_9^\ast$ and $\lambda + \mu \ne 0$. The pairs $\{ \lambda, \mu \}$ and $\{ a\lambda, a\mu \}$ are identified. $x^2 - (\lambda + \mu)x + \lambda\mu$, again with identification. Same as characteristic polynomial. 90 3 270
Total NA NA NA NA 11 720