Survey Article Simple Groups and Simple Lie Algebras

One of the main aims of workers in the theory of groups has always been the determination of all finite simple groups. For simple groups may be regarded as the fundamental building blocks out of which finite groups are constructed. The cyclic groups of prime order are trivial examples of simple groups, and are the only simple groups which are Abelian. The first examples of non-Abelian simple groups were discovered by Galois, who showed that the alternating group An is simple if n ^ 5. The group A5 of order 60 is the smallest non-Abelian simple group. Further examples of finite simple groups are the so-called classical groups, i.e. the linear, symplectic, orthogonal and unitary groups over finite fields, which were first introduced by Jordan [8] and studied in detail by Dickson [4]. These groups are defined as follows. GF(q) denotes the Galois field with q elements, where q is any prime power.