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

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This article gives specific information, namely, element structure, about a particular group, namely: general linear group:GL(2,3).
View element structure of particular groups | View other specific information about general linear group:GL(2,3)

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.

See also element structure of general linear group of degree two.

Conjugacy class structure

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

Compare with element structure of general linear group of degree two#Conjugacy class structure.

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