Questions of this type have been raised about various finite groups of Lie type at MathOverflow previously, for example here. As Nick Gill's comment indicates, the work of E. Vvodin is worth consulting, along with an earlier paper by M. Barry, etc. Naturally the general (or special) linear group over a finite field is somewhat easier to study directly, using a mixture of techniques from linear algebra and finite group theory. But there is some advantage in looking at all finite groups of Lie type from the perspective of algebraic groups.

Two basic questions tend to arise: (1) determine (up to conjugacy) all maximal abelian subgroups, (2) find the largest order of any such subgroup. From either the linear algebra or the algebraic group viewpoint, a natural tool here is the Jordan decomposition of elements. It turns out that semisimple elements (those of order not divisible by $p$) are the easiest to study systematically, largely because their centralizers are again reductive -- and even connected when the algebraic group is simply connected (true here for $\mathrm{SL}_n$).

In particular, an abelian subgroup $A$ of the finite group $G$ consisting of semisimple elements always lies in a maximal torus of the algebraic group defined over $\mathbb{F}_q$. This is one of the results developed for all finite Chevalley groups by Springer and Steinberg in their extensive notes on conjugacy classes: Part E in Seminar on Algebraic Groups and Related Finite Groups (Springer Lecture Notes in Math. 131, 1970), II, 5.8-5.12. (But there are nuances for some primes in types other than the special linear groups.) In particular, the groups of rational points (or fixed points under Steinberg's endomorphism $\sigma$) in the various maximal tori are easily seen to be maximal abelian subgroups and have orders specified in terms of data from the Weyl group, here $S_n$: II, 1.7. These orders are approximately $q^n$ for the finite general linear groups ($n$ being the overall rank). Inductive methods like those suggested by Geoff Robinson are often helpful when only semisimple elements are discussed.

The complication is that the maximum order of an abelian subgroup is approximately $q^{n^2/4}$, typically much larger than a finite torus. Since the centralizers in the algebraic group of nontrivial unipotent elements (= elements having $p$-power orders) are usually far from being reductive, it is tricky to work out the orders of all maximal abelian subgroups of $G$ which involve such elements.

Questions of this type have been raised about various finite groups of Lie type at MathOverflow previously, for example here. As Nick Gill's comment indicates, the work of E. Vvodin is worth consulting, along with an earlier paper by M. Barry, etc. Naturally the general (or special) linear group over a finite field is somewhat easier to study directly, using a mixture of techniques from linear algebra and finite group theory. But there is some advantage in looking at all finite groups of Lie type from the perspective of algebraic groups.

Two basic questions tend to arise: (1) determine (up to conjugacy) all maximal abelian subgroups, (2) find the largest order of any such subgroup. From either the linear algebra or the algebraic group viewpoint, a natural tool here is the Jordan decomposition of elements. It turns out that semisimple elements (those of order not divisible by $p$) are the easiest to study systematically, largely because their centralizers are again reductive -- and even connected when the algebraic group is simply connected (true here for $\mathrm{SL}_n$).

In particular, an abelian subgroup $A$ of the finite group $G$ consisting of semisimple elements always lies in a maximal torus of the algebraic group defined over $\mathbb{F}_q$. This is one of the results developed for all finite Chevalley groups by Springer and Steinberg in their extensive notes on conjugacy classes: Part E in Seminar on Algebraic Groups and Related Finite Groups (Springer Lecture Notes in Math. 131, 1970), II, 5.8-5.12. (But there are nuances for some primes in types other than the special linear groups.) In particular, the groups of rational points (or fixed points under Steinberg's endomorphism $\sigma$) in the various maximal tori are easily seen to be maximal abelian subgroups and have orders specified in terms of data from the Weyl group, here $S_n$: II, 1.7. These orders are approximately $q^n$ for the finite general linear groups ($n$ being the overall rank). Inductive methods like those suggested by Geoff Robinson are often helpful when only semisimple elements are discussed.

The complication is that the maximum order of an abelian subgroup is approximately $q^{n^2/4}$, typically much larger than a finite torus. Since the centralizers in the algebraic group of nontrivial unipotent elements (= elements having $p$-power orders) are usually far from being reductive, it is tricky to work out the orders of all maximal abelian subgroups of $G$ which involve such elements.

Questions of this type have been raised about various finite groups of Lie type at MathOverflow previously, for example here. As Nick Gill's comment indicates, the work of E. Vvodin is worth consulting, along with an earlier paper by M. Barry, etc. Naturally the general (or special) linear group over a finite field is somewhat easier to study directly, using a mixture of techniques from linear algebra and finite group theory. But there is some advantage in looking at all finite groups of Lie type from the perspective of algebraic groups.

Two basic questions tend to arise: (1) determine (up to conjugacy) all maximal abelian subgroups, (2) find the largest order of any such subgroup. From either the linear algebra or the algebraic group viewpoint, a natural tool here is the Jordan decomposition of elements. It turns out that semisimple elements (those of order not divisible by $p$) are the easiest to study systematically, largely because their centralizers are again reductive -- and even connected when the algebraic group is simply connected (true here for $\mathrm{SL}_n$).

In particular, an abelian subgroup $A$ of the finite group $G$ always lies in a maximal torus of the algebraic group defined over $\mathbb{F}_q$. This is one of the results developed for all finite Chevalley groups by Springer and Steinberg in their extensive notes on conjugacy classes: Part E in Seminar on Algebraic Groups and Related Finite Groups (Springer Lecture Notes in Math. 131, 1970), II, 5.8-5.12. (But there are nuances for some primes in types other than the special linear groups.) In particular, the groups of rational points (or fixed points under Steinberg's endomorphism $\sigma$) in the various maximal tori have orders specified in terms of data from the Weyl group, here $S_n$: II, 1.7. These orders are approximately $q^n$ for the finite general linear groups ($n$ being the overall rank). Inductive methods like those suggested by Geoff Robinson are often helpful when only semisimple elements are discussed.

The complication is that the maximum order of an abelian subgroup is approximately $q^{n^2/4}$, typically much larger than a finite torus. Since the centralizers in the algebraic group of nontrivial unipotent elements (= elements having $p$-power orders) are usually far from being reductive, it is tricky to work out the orders of all maximal abelian subgroups of $G$ which involve such elements.

Questions of this type have been raised about various finite groups of Lie type at MathOverflow previously, for example here. As Nick Gill's comment indicates, the work of E. Vvodin is worth consulting, along with an earlier paper by M. Barry, etc. Naturally the general (or special) linear group over a finite field is somewhat easier to study directly, using a mixture of techniques from linear algebra and finite group theory. But there is some advantage in looking at all finite groups of Lie type from the perspective of algebraic groups.

Two basic questions tend to arise: (1) determine (up to conjugacy) all maximal abelian subgroups, (2) find the largest order of any such subgroup. From either the linear algebra or the algebraic group viewpoint, a natural tool here is the Jordan decomposition of elements. It turns out that semisimple elements (those of order not divisible by $p$) are the easiest to study systematically, largely because their centralizers are again reductive -- and even connected when the algebraic group is simply connected (true here for $\mathrm{SL}_n$).

In particular, an abelian subgroup $A$ of the finite group $G$ consisting of semisimple elements always lies in a maximal torus of the algebraic group defined over $\mathbb{F}_q$. This is one of the results developed for all finite Chevalley groups by Springer and Steinberg in their extensive notes on conjugacy classes: Part E in Seminar on Algebraic Groups and Related Finite Groups (Springer Lecture Notes in Math. 131, 1970), II, 5.8-5.12. (But there are nuances for some primes in types other than the special linear groups.) In particular, the groups of rational points (or fixed points under Steinberg's endomorphism $\sigma$) in the various maximal tori are easily seen to be maximal abelian subgroups and have orders specified in terms of data from the Weyl group, here $S_n$: II, 1.7. These orders are approximately $q^n$ for the finite general linear groups ($n$ being the overall rank). Inductive methods like those suggested by Geoff Robinson are often helpful when only semisimple elements are discussed.

The complication is that the maximum order of an abelian subgroup is approximately $q^{n^2/4}$, typically much larger than a finite torus. Since the centralizers in the algebraic group of nontrivial unipotent elements (= elements having $p$-power orders) are usually far from being reductive, it is tricky to work out the orders of all maximal abelian subgroups of $G$ which involve such elements.