Subgroup structure of special linear group:SL(2,9)

This article describes the subgroup structure of special linear group:SL(2,9), which is the special linear group of degree two over field:F9. The group has order 720.

Sylow subgroups
We are considering the group $$SL(2,q)$$ with $$q = p^r$$ a prime power, $$q = 9, p = 3, r = 2$$. The prime $$p = 3$$ is the characteristic prime.

Sylow subgroups for the prime 3
The prime 3 is the characteristic prime $$p$$, so we compare with the general information on $$p$$-Sylow subgroups of $$SL(2,q)$$.

Sylow subgroups for the prime 2
We are in the subcase where $$\ell = 2$$ ($$\ell$$ being the prime for which we are taking Sylow subgroups) and $$q \equiv 1 \pmod 8$$. The value $$t$$ such that $$2^t$$ is the largest power of 2 dividing $$q - 1$$ is $$t = 3$$.

Sylow subgroups for the prime 5
Here, $$\ell = 5$$ and we are interested in the $$\ell$$-Sylow subgroups.

We are in the subcase $$\ell$$ is an odd prime dividing $$q + 1$$. Suppose $$\ell^t$$ is the largest power of $$\ell$$ dividing $$q + 1$$. In our case, $$t = 1$$.