Mathieu Groups
For more details on this topic, see Mathieu group.The group PSL(3,4) can be used to construct the Mathieu group M24, one of the sporadic simple groups; in this context, one refers to PSL(3,4) as M21, though it is not properly a Mathieu group itself. One begins with the projective plane over the field with four elements, which is a Steiner system of type S(2,5,21) – meaning that it has 21 points, each line ("block", in Steiner terminology) has 5 points, and any 2 points determine a line – and on which PSL(3,4) acts. One calls this Steiner system W21 ("W" for Witt), and then expands it to a larger Steiner system W24, expanding the symmetry group along the way: to the projective general linear group PGL(3,4), then to the projective semilinear group PΓL(3,4), and finally to the Mathieu group M24.
M24 also contains copies of PSL(2,11), which is maximal in M22, and PSL(2,23), which is maximal in M24, and can be used to construct M24.
Read more about this topic: Projective Linear Group, Finite Fields
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