Construction and Simplicity of the Large Mathieu Groups

Author

Robert Peter Hansen, San Jose State UniversityFollow

Publication Date

Summer 2011

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

Advisor

Timothy H. Hsu

Keywords

5-transitive, hexad, Mathieu group, octad, Steiner system

Subject Areas

Mathematics

Abstract

In this thesis, we describe the construction of the Mathieu group M₂₄ given by Ernst Witt in 1938, a construction whose geometry was examined by Jacques Tits in 1964. This construction is achieved by extending the projective semilinear group PΓL₃(F₄) and its action on the projective plane P²(F₄). P²(F₄) is the projective plane over the field of 4 elements, with 21 points and 21 lines, and PΓL₃(F₄) is the largest group sending lines to lines in P²(F₄).This plane has 168 6-point subsets, hexads, with the property that no 3 points of a hexad are collinear. Under the action of the subgroup PSL₃(F₄), the hexads in P²(F₄) break into 3 orbits of equal size. These orbits are preserved and permuted by PΓL₃(F₄), and can be viewed as 3 points, which, when added to the 21 points of P²(F₄), yield a set X of 24 points. Using lines and hexads in P²(F₄), we define certain 8-point subsets of X, view them as vectors in F₂²⁴ and define the subspace they span as the Golay 24-code. We then define M₂₄ as the automorphism group of the Golay 24-code and show that it acts 5-transitively on X, establishing its simplicity. We calculate the order of M₂₄ and the order of two simple subgroups, M₂₃and M₂₂, the other large Mathieu groups.

Recommended Citation

Hansen, Robert Peter, "Construction and Simplicity of the Large Mathieu Groups" (2011). Master's Theses. 4053.
DOI: https://doi.org/10.31979/etd.qnhv-a5us
https://scholarworks.sjsu.edu/etd_theses/4053