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

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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