The Known Simple Groups

Abstract

In this chapter we give a brief account of the known simple groups. These include the trivial groups of prime order, the alternating groups of degree at least 5, the groups of Lie type, and the 26 sporadic groups. The groups of Lie type are analogues over finite fields of the complex Lie groups, and they constitute the broadest category of known simple groups. The finite case involves a few more “types” than in the complex case; in particular, the family of Suzuki groups and the two families of Ree have no direct complex analogues. The groups of Lie type will be described in Section 2.1, the balance of the chapter being devoted to the sporadic groups.