Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size
From Groupprops
Statement
Suppose is a natural number. Then, there exists a PORC function
of degree
such that, for any prime power
, the number of conjugacy classes in the projective special linear group
(i.e., the projective special linear group of degree
over the finite field of size
) is
.
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 to define the PORC function. In fact, for a field size of
, the polynomial depends only on the value
.
General observations
Below, we list some general observations about the PORC function in giving number of conjugacy classes in
.
Item | Value |
---|---|
Degree of polynomial | ![]() |
Leading coefficient of polynomial | The leading coefficient of the polynomial for ![]() ![]() |
Factors of polynomial | There are no common factors to all the polynomials |
Coefficients of polynomial | It seems like they are all integer multiples of the leading coefficient |
Comparison between polynomials | Suppose ![]() ![]() ![]() ![]() ![]() In particular, the smallest polynomial corresponds to congruence classes that are 1 more than classes relatively prime to ![]() ![]() NOTE: This behavior is opposite to that of the PORC functions for ![]() ![]() However, if we consider the polynomials obtained by dividing by the leading coefficient to make the polynomials monic, and then compare, the behavior reverses and becomes similar to that for ![]() ![]() |
Particular cases
![]() |
![]() |
Possibilities for ![]() |
Corresponding congruence classes mod ![]() ![]() |
Corresponding polynomials in PORC function of ![]() ![]() |
More information |
---|---|---|---|---|---|
1 | 0 | 1 | 1 | 1 | the group is a trivial group |
2 | 1 | 1 2 |
0 1 |
![]() ![]() |
element structure of projective special linear group of degree two over a finite field |
3 | 2 | 1 3 |
0 or 2 1 |
![]() ![]() |
element structure of projective special linear group of degree three over a finite field |
4 | 3 | 1 2 4 |
0 or 2 3 1 |
![]() ![]() ![]() |
element structure of projective special 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 projective general linear group of fixed degree over a finite field is PORC 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