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

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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.

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

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

Compare with element structure of projective general linear group of degree two#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