Project IV 2026-27

The Sporadic Simple Groups

Superviser: Jamie Mason

Project Research Area: Algebra

Description

We call a finite group simple if has no nontrivial normal subgroups. The Jordan-Hölder theorem gives us a way to describe every finite group in terms of a set of simple groups, so we can consider simple groups as the building blocks of finite groups.

One of the most significant results in finite group theory is the classification of finite simple groups. This states that all finite simple groups belong to one of four families:

  • Cyclic groups \(C_p\) for \(p\) a prime.
  • Alternating groups (which are groups of even permutations \(A_n \trianglelefteq S_n\)) for \(n \geq 5\).
  • Groups of Lie type (these consist of a further 16 infinite families of matrix groups over field of prime power characteristic, examples include projective special linear groups and orthogonal groups).
  • 26 so called "sporadic" groups.
The first three of these families are infinite but the last is of particular interest. The sporadic groups are sorted into the last category simply because these are the simple groups that do not fit into any of the other categories. The first of these groups was described in 1861 and it was not until the mid 80s that the last was found. They arrive in a variety of different ways: as permutation groups, symmetry groups of lattices and subgroups of other sporadic groups.

The purpose of the first part of this project will be to bring you up to speed with some of the basics of group theory that you need, as well as give you an idea of the history of the classification of finite simple groups and the sporadics. You will then pick a sporadic group (or collection of sporadic groups) and investigate them further. This investigation can include different ways to construct your chosen group, showing that it is simple, showing how it relates to other sporadics and discussing its connection to other areas of mathematics.

Mode of operation and evidence of learning

This project will revolve around learning through reading with a focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding.

Students will demonstrate their understanding of the subject matter by solving relevant problems, exploring and constructing examples, investigating theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites and Companion modules

Prerequisites: Algebra II.

Corequisites: Representation Theory IV.

Depending on your choice of sporadic group, Cryptography and Codes III may also be useful, but it is not a compulsory prerequisite.

References

  • Wilson, R. (2009). The Finite Simple Groups / by Robert Wilson. (1st ed. 2009.). Springer London. https://doi.org/10.1007/978-1-84800-988-2
  • Solomon, R., A brief history of the classification of the finite simple groups, Bull. Amer. Math. Soc. 38 (2001), 315-352, DOI: https://doi.org/10.1090/S0273-0979-01-00909-0, Published electronically: March 27, 2001
If you have any further questions about this project please feel free to send me an email (jamie.j.mason@durham.ac.uk) or come by my office (MCS3043).