Subgroup structure of special linear group:SL(2,5)

This article describes the subgroup structure of special linear group:SL(2,5), which is the special linear group of degree two over field:F5. The group has order 120.

Sylow subgroups
We are considering the group $$SL(2,q)$$ with $$q = p^r$$ a prime power, $$q = 5, p = 5, r = 1$$. The prime $$p = 5$$ is the characteristic prime.

Sylow subgroups for the prime 5
The prime 5 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 $$q \equiv 5 \pmod 8$$.

Sylow subgroups for the prime 3
Here, $$\ell = 3$$ 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, $$t = 1$$.