View source for Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size

==Statement==

Suppose <math>n</math> is a [[natural number]]. Then, there exists a [[fact about::PORC function;2| ]][[PORC function]] <math>f</math> of degree <math>n - 1</math> such that, for any [[prime power]] <math>q</math>, the [[fact about::number of conjugacy classes;3| ]][[number of conjugacy classes]] in the [[fact about::projective general linear group;3| ]][[projective general linear group]] <math>PGL(n,q)</math> (i.e., the projective general linear group of degree <math>n</math> over the [[finite field]] of size <math>q</math>) is <math>f(q)</math>. 

A PORC function is a ''polynomial on residue classes'' -- it looks like different polynomial functions on different congruence classes modulo a particular number. In this case, we only need to consider congruence classes modulo <math>n</math> to define the PORC function. In fact, for a field size of <math>q</matH>, the polynomial depends ''only'' on the value <math>\operatorname{gcd}(n,q - 1)</math>.

==General observations==

Below, we list some general observations about the PORC function in <math>q</math> giving number of conjugacy classes in <math>PGL(n,q)</math>.

{| class="sortable" border="1"
! Item !! Value
|-
| Degree of polynomial || <math>n - 1</math> for ''all'' the polynomials
|-
| Leading coefficient of polynomial || 1 for ''all'' the polynomials, i.e., each of the polynomials is monic
|-
| Factors of polynomial || There are no common factors to all the polynomials
|-
| Coefficients of polynomial || Is it also true that all coefficients of the polynomial are integers? (Seems so)<br>Are they always positive? (seems so)
|-
| Comparison between polynomials || Suppose <math>q_1,q_2</math> are prime powers such that <math>\operatorname{gcd}(n,q_1 - 1)</math> divides <math>\operatorname{gcd}(n,q_2 - 1)</math>. Then, the polynomial that works on the residue class of <math>q_2</math> is ''bigger'' than the polynomial that works on the residue class of <math>q_1</math>, where ''bigger'' means that the difference has positive leading coefficient.<br>In particular, the ''smallest'' polynomial corresponds to congruence classes that are 1 more than classes relatively prime to <math>n</math>, and the ''largest'' polynomial corresponds to the congruence class 1 mod <math>n</math>.
|}

==Particular cases==

{| class="sortable" border="1"
! <math>n</math> (degree of projective general linear group, also modulus to which we need to consider congruence classes) !! <math>n - 1</math> (degree of PORC function) !! Possibilities for <math>\operatorname{gcd}(n,q - 1)</math> !! Corresponding congruence classes mod <math>n</math> for <math>q</math> !! Corresponding polynomials in PORC function of <math>q</math> giving number of conjugacy classes in <math>PGL(n,q)</math> !! More information
|-
| 1 || 0 || 1 || 1 || 1 || the group is a [[trivial group]]
|-
| 2 || 1 || 1<br>2 || 0<br>1 || <math>q + 1</math><br><math>q + 2</math> || See [[element structure of projective general linear group of degree two over a finite field]]
|-
| 3 || 2 || 1<br>3 || 0 or 2<br>1 || <math>q^2 + q</math><br><math>q^2 + q + 2</math>|| See [[element structure of projective general linear group of degree three over a finite field]]
|-
| 4 || 3 || 1<br>2<br>4 || 0 or 2<br>3<br>1 || <math>q^3 + q^2 + q</math><br><math>q^3 + q^2 + 2q + 1</math><br><math>q^3 + q^2 + 2q + 3</math> || [[element structure of projective general linear group of degree four over a finite field]]
|}

==Related facts==

* [[Number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size]]
* [[Number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size]]
* [[Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size]]