Finite Classical Groups

This repository consists of lecture notes for five talks on finite classical groups which I gave as part of a reading course in the Spring semester of 2026 at the University of Ljubljana, Faculty of Mathematics and Physics (UL FMF).

The course was supervised by Prof. Dr. Primož Potočnik.

Overview

These notes provide a concise introduction to the geometry and structure of the finite classical groups. The primary focus is on the definition of these groups and the proof of their simplicity using Iwasawa’s Criterion.

Lecture Notes

The following PDFs summarize the five lectures of the course:

• Lecture 1: Linear Groups – Semilinear transformations, $PSL_n(q)$, and Iwasawa’s Criterion.
• Lecture 2: Forms – Sesquilinear and quadratic forms, Witt’s Lemma, and the geometry of forms.
• Lecture 3: Symplectic Groups – Structure and properties of $PSp_{2n}(q)$.
• Lecture 4: Unitary Groups – The structure of $PSU_n(q^2)$.
• Lecture 5: Orthogonal Groups – The structure of $P\Omega_n^{\pm}(q)$.

References

The material is primarily based on the following sources:

1. N. Gill, Finite Classical Groups, LTCC Lecture Notes, 2025.
2. P. J. Cameron, Notes on Classical Groups, Queen Mary and Westfield College, London, 2000.
3. P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge University Press, 1990.

About the Author

Matevž Miščič
PhD Student at UL FMF
Research Interests: Babai’s Conjecture, Diameters of Groups, and Lie Algebras.

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Note: These notes were prepared for personal study. While every effort has been made to ensure accuracy, they are provided “as is”.