Element structure of projective general linear group of degree three over a finite field
This article gives specific information, namely, element structure, about a family of groups, namely: projective general linear group of degree three.
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We take as the number of elements in the field and as the underlying prime number, so is a power of .
- Element structure of special linear group of degree three over a finite field
- Element structure of general linear group of degree three over a finite field
- Element structure of projective special linear group of degree three over a finite field
|number of conjugacy classes|| Case is not 1 mod 3 (e.g., ): |
Case (e.g., ):
Conjugacy class structure
Number of conjugacy classes
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 3 - 1 = 2, and we need to make cases based on the congruence class of the field size modulo 3. 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 mod 3||Number of conjugacy classes (polynomial of degree 3 - 1 = 2 in )||Additional comments|
|1|| 0 or -1 mod 3:
0 mod 3 (e.g. )
-1 mod 3 (e.g., )
|In this case, we have an isomorphism between linear groups when degree power map is bijective, so|
|3||1 mod 3 (e.g., )|