Suppose is a field of size , where is a power of an odd prime . Suppse is a generator of the multiplicative group of . Consider the subgroup of the general linear group given by:
Then, one of these cases holds:
- , i.e., is the entire special linear group.
- , and is a quasisimple group of order , with isomorphic to the alternating group of degree five.
In either case, contains a subgroup isomorphic to SL(2,3).
- upper-triangular and lower-triangular unipotent matrices generate free non-abelian subgroup in special linear group over integers: In , picking matrices of the above kind yields a free non-abelian subgroup with those matrices as generators. This follows from the ping-pong lemma.