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

This article gives specific information, namely, element structure, about a family of groups, namely: projective general linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.
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This article describes the element structure of the projective general linear group of degree two over a finite field of order $q$ (a prime power) and characteristic $p$ (a prime number). $q$ is a power of $p$. We denote by $r$ the positive integer value $\log_pq$, so $q = p^r$. Some aspects of this discussion, with suitable infinitary analogues of cardinality, carry over to infinite fields and fields of infinite characteristic.

Related information

Generalizing to infinite fields and non-fields

• Element structure of projective general linear group of degree two over a field
• Element structure of projective general linear group of degree two over a finite discrete valuation ring

Related groups over finite fields

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

Summary

Item	Value
order of the group	$q^3 - q = q(q - 1) = q(q - 1)(q + 1)$
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)	Case on $q$	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	even	symmetric group:S3	6	1,2,3	3	element structure of symmetric group:S3
3	3	odd	symmetric group:S4	24	1,3,6,6,8	5	element structure of symmetric group:S4
4	2	even	alternating group:A5	60	1,12,12,15,20	5	element structure of alternating group:A5
5	5	odd	symmetric group:S5	120	1,10,15,20,20,24,30	7	element structure of symmetric group:S5
7	7	odd	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	even	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	odd	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	$q + 1$	$q^3 - q$