Splitting criterion for conjugacy classes in special linear group of prime degree over a finite field

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Statement

Suppose n is a prime number and q is a prime power (the underlying prime of q may or may not be equal to n). Then, there is a finite field \mathbb{F}_q of size q, unique up to isomorphism. We denote the special linear group SL(n,\mathbb{F}_q) by SL(n,q). Similarly, we denote the general linear group GL(n,\mathbb{F}_q) by GL(n,q).

Our goal is to determine the splitting criterion for elements between GL(n,q) and the normal subgroup SL(n,q). More explicitly, we are interested in the question: given an element g \in SL(n,q), under what conditions is the GL(n,q)-conjugacy class of g the same as its SL(n,q)-conjugacy class? Further, if they are not equal, how many different SL(n,q)-conjugacy classes does the GL(n,q)-conjugacy class split into?

Case of no nth roots of unity

If \operatorname{gcd}(n,q - 1) = 1 (this is the same as saying that q is not 1 mod n), then none of the conjugacy classes split. In fact, in this case, SL(n,q) is a direct factor of GL(n,q), 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 \operatorname{gcd}(n,q - 1) = n (this is the same as saying that q is 1 mod n), then the only conjugacy classes that split are the ones that comprise a single Jordan block of size n:

  • There are n such GL(n,q)-conjugacy classes in SL(n,q). These correspond to eigenvalue choices that are the n different n^{th} roots of unity.
  • Each such GL(n,q)-conjugacy class splits into n conjugacy classes in SL(n,q).
  • Thus, there is a total of n^2 conjugacy classes in SL(n,q) that we obtain after the splitting.

Particular cases

Value of prime n Element structure of SL(n,q) for general q
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