Master of Science (MS)
Timothy H. Hsu
5-transitive, hexad, Mathieu group, octad, Steiner system
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.
Hansen, Robert Peter, "Construction and Simplicity of the Large Mathieu Groups" (2011). Master's Theses. 4053.