SIMPLICITY OF An

A finite group is called simple when its only normal subgroups are the trivial subgroup and the whole group. For instance, a finite group of prime size is simple, since it in fact has no non-trivial proper subgroups at all (normal or not). A finite abelian group G not of prime size, is not simple: let p be a prime factor of #G, so G contains a subgroup of order p, which is a normal since G is abelian and is proper since #G > p. Thus, the abelian finite simple groups are the groups of prime size. When n ≥ 3 the group Sn is not simple because it has a nontrivial normal subgroup An. But the groups An are simple, provided n ≥ 5. Theorem 1.1 (C. Jordan, 1875). For n ≥ 5, the group An is simple. The restriction n ≥ 5 is optimal, since A4 is not simple: it has a normal subgroup of size 4, namely {(1), (12)(34), (13)(24), (14)(23)}. The group A3 is simple, since it has size 3, and the groups A1 and A2 are trivial. We will give five proofs of Theorem 1.1. Section 2 includes some preparatory material and later sections give the proofs of Theorem 1.1. In the final section, we give a quick application of the simplicity of alternating groups and give references for further proofs not treated here.