2 - Mathieu groups

Summary

In this chapter we construct the Mathieu groups and study their basic properties. We construct the largest Mathieu group Mat₂₄ as the automorphism group of the (binary extended) Golay code defined in Section 2.1. In Section 2.2 we construct a Golay code as the quadratic residue code over GF(23). In Section 2.3 we show that a minimal non-empty subset in a Golay code has size 8 (called an octad). Moreover the set of all octads in a Golay code forms the block set of a Steiner system of type S(5,8,24). The residue of a 3-element subset of elements in a Steiner system of type S(5,8,24) is a projective plane of order 4. In Section 2.4 we review some basic properties of the linear groups and in Sections 2.5 and 2 6 we define the generalized quadrangle of order (2,2) and its triple cover which is the tilde geometry of rank 2. In Section 2.7 we prove uniqueness of the projective plane of order 4 and analyse some properties of the plane and its automorphism group. This analysis enables us to establish the uniqueness of the Steiner system of type S(5,8,24) in Section 2.8. The Mathieu group Mat₂₄ of degree 24 is defined in Section 2.9 as the automorphism group of the unique Golay code. The uniqueness proof implies rather detailed information about Mat₂₄ and two other large Mathieu groups Mat₂₃ and Mat₂₂. In Section 2.10 we study the stabilizers in Mat₂₄ of an octad, a trio and a sextet.