Splitting criterion for conjugacy classes in special linear group of prime degree over a finite field
Suppose is a prime number and is a prime power (the underlying prime of may or may not be equal to ). Then, there is a finite field of size , unique up to isomorphism. We denote the special linear group by . Similarly, we denote the general linear group by .
Our goal is to determine the splitting criterion for elements between and the normal subgroup . More explicitly, we are interested in the question: given an element , under what conditions is the -conjugacy class of the same as its -conjugacy class? Further, if they are not equal, how many different -conjugacy classes does the -conjugacy class split into?
Case of no nth roots of unity
If (this is the same as saying that is not 1 mod ), then none of the conjugacy classes split. In fact, in this case, is a direct factor of , hence a conjugacy-closed subgroup; for more, see isomorphism between linear groups when degree power map is bijective.
Case of primitive nth roots of unity
If (this is the same as saying that is 1 mod ), then the only conjugacy classes that split are the ones that comprise a single Jordan block of size :
- There are such -conjugacy classes in . These correspond to eigenvalue choices that are the different roots of unity.
- Each such -conjugacy class splits into conjugacy classes in .
- Thus, there is a total of conjugacy classes in that we obtain after the splitting.
|Value of prime||Element structure of for general|
|2||element structure of special linear group of degree two over a finite field|
|3||element structure of special linear group of degree three over a finite field|