Projective Linear Group - Finite Fields - Mathieu Groups

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.