Subgroup structure of special linear group of degree two over a finite field

Sylow subgroups
We consider the group $$SL(2,q)$$ over the field $$\mathbb{F}_q$$ of $$q$$ elements. $$q$$ is a prime power of the form $$p^r$$ where $$p$$ is a prime number and $$r$$ is a positive integer. $$p$$ is hence also the characteristic of $$\mathbb{F}_q$$. We call $$p$$ the characteristic prime.

Sylow subgroups for other primes: cases and summary
For any prime $$\ell$$, the $$\ell$$-Sylow subgroup is nontrivial iff $$\ell \mid q^3 - q$$. If $$\ell \ne p$$, then it does not divide $$q$$, so we get that $$\ell \mid q^2 - 1$$ which means that either $$\ell \mid q - 1$$ or $$\ell \mid q + 1$$. Further, if $$\ell \ne 2$$, exactly one of these cases can occur. For $$\ell = 2$$, we make cases based on the residue of $$q$$ mod 8. The summary of cases is below and more details are in later sections.

Sylow subgroups for odd primes dividing $$q - 1$$
Suppose $$\ell$$ is an odd prime dividing $$q - 1$$. Note that $$\ell \ne p$$ and $$\ell$$ does not divide $$q + 1$$. Suppose $$\ell^t$$ is the largest power of $$\ell$$ dividing $$q - 1$$.

Sylow subgroups for odd primes dividing $$q + 1$$
Suppose $$\ell$$ is an odd prime dividing $$q + 1$$. Note that $$\ell \ne p$$ and $$\ell$$ does not divide $$q - 1$$. Suppose $$\ell^t$$ is the largest power of $$\ell$$ dividing $$q + 1$$.

Sylow subgroups for the prime two where the field size is 1 mod 8
In this case, $$(q + 1)/2$$ is odd whereas $$(q - 1)/2$$ is even. In fact, $$(q -1)/4$$ is also even. Let $$t$$ be such that $$2^t$$ is the largest power of 2 dividing $$q - 1$$. Note that $$t \ge 3$$.

Sylow subgroups for the prime two where the field size is 5 mod 8
In this case, $$(q + 1)/2$$ is odd whereas $$(q - 1)/2$$ is even. However, $$(q - 1)/4$$ is odd. Then, $$2^1$$ is the largest power of 2 dividing $$q + 1$$ and $$2^2$$ is the largest power of 2 dividing $$q - 1$$.

Sylow subgroups for the prime two where the field size is 7 mod 8
In this case, $$(q - 1)/2$$ is odd whereas $$(q + 1)/2$$ is even. Let $$t$$ be such that $$2^t$$ is the largest power of 2 dividing $$q + 1$$. Note that $$t \ge 3$$.

Sylow subgroups for the prime two where the field size is 3 mod 8
In this case, $$(q - 1)/2$$ is odd whereas $$(q + 1)/2$$ is even, but $$(q + 1)/4$$ is odd. Then, the largest power of 2 dividing $$q - 1$$ is 2 and the largest power of 2 dividing $$q + 1$$ is 4.