Maximal subgroups of finite simple groups: classifications and applications

This paper surveys what is currently known about the maximal subgroups of the finite simple groups. After briefly introducing the groups themselves, if their maximal subgroups are completely determined then we present this classification. For the remaining finite simple groups our current knowledge is only partial: we describe the state of play, as well as giving some results that apply more generally. We also direct the reader towards computational resources for the construction of maximal subgroups. After this, we present three sample applications, selected because they combine group theoretical and combinatorial arguments, and because they use either or both of the detailed classifications and the looser statements that can be made about all maximal subgroups. In particu- lar, we discuss results relating to generation, and the generating graph; results concerning bases; and some applications to computational com- plexity, in particular to graph colouring and other problems with no known polynomial-time solution. Postprint