Symmetric and Alternating Groups
We have seen the definition of the symmetric group \(S_n\), but so far, we do not have too much experience with it. In this chapter, we will introduce the notions of cycles and, in particular, transpositions, which are important elements of the symmetric group. These will help us to understand the group. We will also construct a subgroup of the symmetric group called the alternating group. If \(n\ge 5\), then the alternating group is very special in that it has no nontrivial proper normal subgroups.