Editing Projective linear group (section)

=== Mathieu groups ===
{{details|Mathieu group}}
The group {{nowrap|PSL(3, 4)}} can be used to construct the [[Mathieu group]] M<sub>24</sub>, one of the [[sporadic simple group]]s; in this context, one refers to {{nowrap|PSL(3, 4)}} as M<sub>21</sub>, 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 {{nowrap|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 {{nowrap|PSL(3, 4)}} acts. One calls this Steiner system W<sub>21</sub> ("W" for [[Ernst Witt|Witt]]), and then expands it to a larger Steiner system W<sub>24</sub>, expanding the symmetry group along the way: to the projective general linear group {{nowrap|PGL(3, 4)}}, then to the [[projective semilinear group]] {{nowrap|PΓL(3, 4)}}, and finally to the Mathieu group M<sub>24</sub>.

M<sub>24</sub> also contains copies of {{nowrap|PSL(2, 11)}}, which is maximal in M<sub>22</sub>, and {{nowrap|PSL(2, 23)}}, which is maximal in M<sub>24</sub>, and can be used to construct M<sub>24</sub>.<ref>Conway, Sloane, SPLAG</ref>