Publication Date

Summer 2011

Degree Type


Degree Name

Master of Science (MS)




Timothy H. Hsu


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

Subject Areas



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.