Dickson's theorem

Statement
Suppose $$F$$ is a field of size $$q$$, where $$q$$ is a power of an odd prime $$p$$. Suppse $$a$$ is a generator of the multiplicative group of $$F$$. Consider the subgroup $$L$$ of the general linear group $$GL(2,F)$$ given by:

$$L = \left \langle \begin{pmatrix}1 & 1 \\ 0 & 1\\\end{pmatrix}, \begin{pmatrix}1 & 0\\a & 1\end{pmatrix}\right \rangle$$.

Then, one of these cases holds:


 * $$L = SL(2,F)$$, i.e., $$L$$ is the entire special linear group.
 * $$q = 9$$, and $$L$$ is a quasisimple group of order $$120$$, with $$L/Z(L)$$ isomorphic to the alternating group of degree five.

In either case, $$L$$ contains a subgroup isomorphic to SL(2,3).

Related facts

 * upper-triangular and lower-triangular unipotent matrices generate free non-abelian subgroup in special linear group over integers: In $$SL(2,\mathbb{Z})$$, picking matrices of the above kind yields a free non-abelian subgroup with those matrices as generators. This follows from the ping-pong lemma.