# 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).

## 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