# Element structure of general linear group of degree three over a finite field

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: general linear group of degree three.
View element structure of group families | View other specific information about general linear group of degree three

This article discusses the element structure of the general linear group of degree three over a finite field. The group is $GL(3,q)$ where $q$ is the order (size) of the field. We denote by $p$ the prime number that is the characteristic of the field.

## Particular cases

Group $p$ $q$ Order of the group Number of conjugacy classes Element structure page
projective special linear group:PSL(3,2) 2 2 168 6 element structure of projective special linear group:PSL(3,2)
general linear group:GL(3,3) 3 3 11232 24 element structure of general linear group:GL(3,3)
general linear group:GL(3,5) 5 5 1488000 120 element structure of general linear group:GL(3,5)

## Conjugacy class structure

There is a total of $(q^3 - 1)(q^3 - q)(q^3 - q^2) = q^3(q - 1)^3(q + 1)(q^2 + q + 1)$ elements, and a total of $q^3 - q = q(q - 1)(q + 1)$ conjugacy classes.

Nature of conjugacy class Eigenvalues Characteristic polynomial Minimal polynomial Size of conjugacy class Number of such conjugacy classes Total number of elements Semisimple? Diagonalizable over $\mathbb{F}_q$?
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar $\{a,a,a \}$ where $a \in \mathbb{F}_q^\ast$ $(x - a)^3$ $x - a$ 1 $q - 1$ $q - 1$ Yes Yes
Diagonalizable over $\mathbb{F}_q$ with one eigenvalue having multiplicity two, the other eigenvalue having multiplicity one $\{ a,a,b \}$ where $a \ne b$, both in $\mathbb{F}_q^\ast$ $(x - a)^2(x - b)$ $(x - a)(x - b)$ $q^2(q^2 + q + 1) = q^4 + q^3 + q^2$ $(q - 1)(q - 2)$ $q^2(q^2 + q + 1)(q - 1)(q - 2) = q^6 - 2q^5 - q^3 + 2q^2$ Yes Yes
Diagonalizable over $\mathbb{F}_q$ with all distinct diagonal entries $\{ a,b,c \}$, all distinct elements of $\mathbb{F}_q^\ast$ $(x - a)(x - b)(x - c)$ same as characteristic polynomial $q^3(q + 1)(q^2 + q + 1)$ $(q-1)(q-2)(q-3)/6 = (q^3 - 6q^2 + 11q - 6)/6$ $q^3(q + 1)(q^2 + q + 1)(q - 1)(q - 2)(q - 3)/6 = (q^9 - 4q^8 + q^7 + 5q^6 + 4q^5 - q^4 - 6q^3)/6$ Yes Yes
Diagonalizable over $\mathbb{F}_{q^3}$, not over $\mathbb{F}_q$ Distinct Galois conjugate triple of elements in $\mathbb{F}_{q^3}^\ast$. If one of the elements is $a$, the other two are $a^q$ and $a^{q^2}$. irreducible degree three polynomial over $\mathbb{F}_q$ same as characteristic polynomial $q^3(q - 1)^2(q + 1)$ $q(q + 1)(q - 1)/3 = (q^3 - q)/3$ $q^4(q - 1)^3(q + 1)^2/3 = (q^9 - q^8 - 2q^7 + 2q^6 + q^5 - q^4)/3$ Yes No
One eigenvalue is in $\mathbb{F}_q^\ast$, the other two are in $\mathbb{F}_{q^2} \setminus \mathbb{F}_q$ one element of $\mathbb{F}_q^\ast$, pair of Galois conjugates over $\mathbb{F}_q$ in $\mathbb{F}_{q^2}$. product of linear polynomial and irreducible degree two polynomial over $\mathbb{F}_q$ same as characteristic polynomial $q^3(q - 1)(q^2 + q + 1) = q^6 - q^3$ $q(q - 1)^2/2 = (q^3 - 2q^2 + q)/2$ $q^4(q - 1)^3(q^2 + q + 1)/2 = (q^9 - 2q^8 + q^7 - q^6 + 2q^5 - q^4)/2$ Yes No
Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over $\mathbb{F}_q$ $\{ a,a,b \}$ with $a,b \in \mathbb{F}_q^\ast$, $a \ne b$ $(x - a)^2(x - b)$ same as characteristic polynomial $q^2(q + 1)(q - 1)(q^2 + q + 1)$ $(q - 1)(q - 2)$ $q^2(q -1)^2(q + 1)(q^2 + q + 1)(q - 2) = q^8 - 2q^7 - q^6 + q^5 + 2q^4 + q^3 - 2q^2$ No No
Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over $\mathbb{F}_q$ $\{ a,a,a \}$ with $a \in \mathbb{F}_q^\ast$ $(x - a)^3$ $(x - a)^2$ $q(q + 1)(q - 1)^2(q^2 + q + 1) = q^6 - q^4 - q^2 + q$ $q - 1$ $q(q + 1)(q - 1)^3(q^2 + q + 1) = q^7 - q^6 - q^5 + q^3 + q^2 - q$ No No
Has Jordan block of size 3 $\{ a,a,a \}$ with $a \in \mathbb{F}_q^\ast$ $(x - a)^3$ same as characteristic polynomial $(q - 1)(q + 1)(q^2 + q + 1) = q^4 + q^3 - q - 1$ $q - 1$ $(q - 1)^2(q + 1)(q^2 + q + 1) = q^5 - q^3 - q^2 + 1$ No No
Total NA NA NA NA $q^3 - q$ $(q^3 - 1)(q^3 - q)(q^3 - q^2) = q^9 - q^8 - q^7 + q^5 + q^4 - q^3$ NA NA