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

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