Dickson's theorem

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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:

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

Related facts

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