Finite groups and geomtries

In finite groups theory, one of the most important studies is examination of finite and simple groups. Many studies have been done so far. Especially, wide researches have been done about Mathieu groups geometries, geometries which belongs different ranks, have been constructed. While doing this, it has been benefitted from subgroups of Mathieu groups. In section 3, we describe the construction of Mathieu groups M9 ,M10 , M11 and M12. After making defination of incidence geometry, the groups of M9 ,M10 , M11 and M12 groups belonging to different ranks of geometries have been examined and incidence geometry diagram has been constructed.