This report documents a winter reading project on the Large Mathieu Groups ($M_{22}$, $M_{23}$, $M_{24}$), which belong to the sporadic simple groups, one of the most exotic families of finite simple groups.
Project Overview
Working through Robert Peter Hanson’s thesis, I explored the construction of these groups via:
- Finite field theory and Galois extensions
- Linear and semi-linear groups over $\mathbb{F}_4$
- Projective geometry in $\mathbb{P}^2(\mathbb{F}_4)$
- Hexads and $k$-arcs in the projective plane
- The 24-point Golay code $C_{24}$
The Mathieu group $M_{24}$ emerges as the automorphism group of the Golay code, acting 5-transitively on 24 points. Its point stabilizers form a descending chain leading to $PSL_3(\mathbb{F}_4)$, whose simplicity can be proven through commutator subgroup analysis.
Key Results Studied
- Construction of $\mathbb{P}^2(\mathbb{F}_4)$ and its group actions
- Classification of hexad orbits under $PSL_3(\mathbb{F}_4)$
- Generation of the Golay code from line and oval octads
- Proofs of simplicity for $M_{22}$, $M_{23}$, and $M_{24}$
The project provided valuable exposure to the interplay between coding theory, finite geometry, and group theory.