Mathieu Groups

Abstract

The Classification of Finite Simple Groups was a prominent goal of algebraists. The Classification Theorem was complete in 1983 and many textbooks from the 1980s include detailed proofs and explorations of many aspects of this subject. For example, J.J. Rotman devotes an entire chapter to the Mathieu groups [13]. It seems that there is still disagreement amongst mathematicians as to whether the Classification Theorem should be deemed thorough or without major error. Looking into the entire Classification Theorem would be a huge undertaking, so in this paper we are discussing only the five sporadic Mathieu groups. Looking at these small sporadic simple groups opened up a discussion of transitivity and k-transitivity. In addition to traditional abstract algebra material, this paper explores a relationship between the five sporadic Mathieu groups and the combinatorial Steiner Systems. Included in this discussion is the relationship of M₂₄ with the Binary Golay Code. This thesis ends in a proof of the simplicity of the Mathieu Groups. The proof of the simplicity of M₁₁ and M₂₃ which was developed by R. Chapman in his note, An elementary proof of the Mathieu groups M₁₁ and M₂₃ makes the preliminary theorems to the simplicity proof in J.J. Rotman's book look much less perfunctory [2],[13]. This raises the question of whether there could possibly be a more succinct proof of the simplicity of M₁₂, M₂₂ and M₂₄.