• The finite simple groups, namely those which have no nontrivial normal subgroups, are the building blocks for all finite groups. Finite simple groups were classified completely in the 1980s: this classification states that all finite simple groups belong to the following list:

  • cyclic groups of prime order
  • alternating groups
  • finite groups of Lie type
  • 26 sporadic groups

Many questions concerning finite groups, their permutation actions and linear representations, can be reduced to questions about simple groups. Thus it is important to have a deep understanding of properties of the finite simple groups. I am particularly interested in the subgroup structure of these groups. The bulk of the finite simple groups are the groups of Lie type, and for these groups there are fruitful ways of applying the work on algebraic groups outlined above. Many results have been proved in this area; in particular, our knowledge of the maximal subgroups of the finite simple groups is now very substantial.