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.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 44, Theorem 8.4, Chapter 2 (Some basic topics), Section 2.8 (Two-dimensional linear and projective groups), ^{More info}