Element structure of projective general linear group of degree two over a finite field

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This article gives the element structure of the projective general linear group of degree two over a finite field. Some of the structure information generalizes to infinite fields.

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

We denote by q the order (size) of the field and by p the prime number that is the characteristic. Note that q must be a power of p.

The order of the group is q^3 - q.

Summary

Item Value
conjugacy class sizes Case q odd (e.g., q = 3,5,7,9,11,13,17): 1 (1 time), q(q-1)/2 (1 time), q(q+1)/2 (1 time), q(q-1) ((q-1)/2 times), q^2 - 1 (1 time), q(q+1) ((q-3)/2 times)
Case q even (e.g., q = 2,4,8,16,32): 1 (1 time), q(q-1) (q/2 times), q^2 - 1 (1 time), q(q+1) ((q - 2)/2 times)
number of conjugacy classes Case q odd: q + 2, Case q even: q + 1
equals number of conjugacy classes, see also linear representation theory of projective general linear group of degree two over a finite field#Conjugacy class structure
number of p-regular conjugacy classes, where p is the field characteristic (so q is a power of p) Case q odd: q + 1, Case q even: q

Particular cases

Field size q Underlying prime p (field characteristic) Group PGL(2,q) Order of the group (= q^3 - q) List of conjugacy class sizes (ascending order) Number of conjugacy classes (= q + 2 if q odd, q + 1 if p = 2) Element structure page
2 2 symmetric group:S3 6 1,2,3 3 element structure of symmetric group:S3
3 3 symmetric group:S4 24 1,3,6,6,8 5 element structure of symmetric group:S4
4 2 alternating group:A5 60 1,12,12,15,20 5 element structure of alternating group:A5
5 5 symmetric group:S5 120 1,10,15,20,20,24,30 7 element structure of symmetric group:S5
7 7 projective general linear group:PGL(2,7) 336 1,21,28,42,42,42,48,56,56 9 element structure of projective general linear group:PGL(2,7)
8 2 projective special linear group:PSL(2,8) 504 1,56,56,56,56,63,72,72,72 9 element structure of projective special linear group:PSL(2,8)
9 3 projective general linear group:PGL(2,9) 720 1,36,45,72,72,72,72,80,90,90,90 11 element structure of projective general linear group:PGL(2,9)

Conjugacy class structure

As we know in general, number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size, the degree of this PORC function is one less than the degree of matrices, and we make cases based on the congruence classes modulo the degree of matrices. Thus, we expect that the number of conjugacy classes is a PORC function of the field size of degree 2 - 1 = 1, and we need to make cases based on the congruence class of the field size modulo 2. Moreover, the general theory also tells us that the polynomial function of q depends only on the value of \operatorname{gcd}(n,q-1), which in turn can be determined by the congruence class of q mod n (with n = 2 here).

Value of \operatorname{gcd}(2,q-1) Corresponding congruence classes of q mod 2 Number of conjugacy classes (polynomial of degree 2 - 1 = 1 in q) Additional comments
1 0 mod 2 (e.g., q = 2,4,8,\dots) q + 1 In this case, we have an isomorphism between linear groups when degree power map is bijective, so SL(2,q) \cong PGL(2,q) \cong PSL(2,q)
2 1 mod 2 (e.g., q = 3,5,7,\dots) q + 2

Summary for odd characteristic p, field size q

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

Summary for p = 2

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 \mathbb{F}_q with equal diagonal entries, hence a scalar. \{ a,a \} for a \in \mathbb{F}_q^\ast, all identified. (x - a)^2 x - a 1 1 1
Diagonalizable over \mathbb{F}_{q^2}, not \mathbb{F}_q. Hence, distinct eigenvalues. Distinct elements of \mathbb{F}_{q^2} x^2 - ax + b, irreducible Same as characteristic polynomial q(q - 1) q/2 q^2(q - 1)/2 = (q^3 - q^2)/2
has Jordan block of size two over \mathbb{F}_q a (multiplicity 2), for a \in \mathbb{F}_q^\ast (x - a)^2 (x - a)^2 q^2 - 1 1 q^2 - 1
Diagonalizable over \mathbb{F}_q with distinct diagonal entries \lambda, \mu distinct nonzero elements x^2 - (\lambda + \mu)x + (\lambda \mu) Same as characteristic polynomial q(q + 1) (q - 2)/2 q(q + 1)(q - 2)/2 = (q^3 - q^2 - 2q)/2
Total NA NA NA NA q + 1 q^3 - q