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 (a prime power) and characteristic
(a prime number).
is a power of
. We denote by
the positive integer value
, so
. 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 | ![]() |
conjugacy class sizes | Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
number of conjugacy classes | Case ![]() ![]() ![]() ![]() 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 ![]() ![]() ![]() ![]() |
Case ![]() ![]() ![]() ![]() |
Particular cases
Field size ![]() |
Underlying prime ![]() |
Case on ![]() |
Group ![]() |
Order of the group (= ![]() |
List of conjugacy class sizes (ascending order) | Number of conjugacy classes (= ![]() ![]() ![]() ![]() |
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 depends only on the value of
, which in turn can be determined by the congruence class of
mod
(with
here).
Value of ![]() |
Corresponding congruence classes of ![]() |
Number of conjugacy classes (polynomial of degree 2 - 1 = 1 in ![]() |
Additional comments |
---|---|---|---|
1 | 0 mod 2 (e.g., ![]() |
![]() |
In this case, we have an isomorphism between linear groups when degree power map is bijective, so ![]() |
2 | 1 mod 2 (e.g., ![]() |
![]() |
Summary for odd characteristic
, field size 
Nature of conjugacy class upstairs in ![]() |
Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
1 | 1 | 1 |
Diagonalizable over ![]() ![]() |
Pair of mutually negative conjugate elements of ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
1 | ![]() |
Diagonalizable over ![]() |
![]() |
![]() |
Same as characteristic polynomial | ![]() |
1 | ![]() |
Diagonalizable over ![]() ![]() |
Pair of conjugate elements of ![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
Not diagonal, has Jordan block of size two | ![]() |
![]() |
Same as characteristic polynomial | ![]() |
1 | ![]() |
Diagonalizable over ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() |
Same as characteristic polynomial. | ![]() |
![]() |
![]() |
Total | NA | NA | NA | NA | ![]() |
![]() |
Summary for 
Nature of conjugacy class upstairs in ![]() |
Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() |
![]() |
1 | 1 | 1 |
Diagonalizable over ![]() ![]() |
Distinct elements of ![]() |
![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
has Jordan block of size two over ![]() |
![]() ![]() |
![]() |
![]() |
![]() |
1 | ![]() |
Diagonalizable over ![]() |
![]() |
![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
Total | NA | NA | NA | NA | ![]() |
![]() |