The alternating groups

Abstract

The most familiar of the (finite non-abelian) simple groups are the alternating groups A _n , which are subgroups of index 2 in the symmetric groups S _n . In this chapter our main aims are to define these groups, prove they are simple, determine their outer automorphism groups, describe in general terms their subgroups, and construct their covering groups. At the end of the chapter we briefly introduce reflection groups as a generalisation of the symmetric groups, as they play an important role not only in the theory of groups of Lie type, but also in the construction of many sporadic groups, as well as in the elucidation of much exceptional behaviour of low-dimensional classical groups.