Mathieu group $M_{24}$

Introduction:

The Mathieu groups are $M_{11}, M_{12}, M_{22}, M_{23}$ and $M_{24},$ which are one family of the $26$ sporadic finite simple groups. They were discovered by the French mathematician Emile Mathieu. Furthermore, the first expression of simplicity and uniqueness of Mathieu groups was in 1930’s in a paper by Witt, and a Steiner system was describled in this project. Now, we are normally using the system to describe these groups. The largest Mathieu group is $M_{24}$ which is $5$-transitive of $24$-point, and it could be defined as a group of preserved permutations of Steiner system $S(5, 8, 24)$. Notice that some of $M_{24}$’s simple subgroups are $M_{11}, M_{12}, M_{22},$ and $M_{23}$. This project start by introducing some basic facts, moving to the definition of Steiner system and some of its properties, followed by discussing the Steiner system $S(5, 8, 24)$, and then presenting the Miracle Octad Generator (MOG). Then we will end by introducing the concept of $M_{24}$.