Notes on the symmetric group

Recall that, given a set X, the set SX of all bijections from X to itself (or, more briefly, permutations of X) is group under function composition. In particular, for each n ∈ N, the symmetric group Sn is the group of permutations of the set {1, . . . , n}, with the group operation equal to function composition. Thus Sn is a group with n! elements, and it is not abelian if n ≥ 3. If X is a finite set with #(X) = n, then any labeling of the elements of X as x1, . . . , xn defines an isomorphism from SX to Sn (see the homework for a more precise statement). We will write elements of Sn using Greek letters σ, τ, ρ, . . . , we denote the identity function by 1, and we just use the multiplication symbol · or juxtaposition to denote composition (instead of the symbol ◦). Recall however that functions work from right to left: thus στ(i) = σ(τ(i)), in other words evaluate first τ at i, and then evaluate σ on this. We say that σ moves i if σ(i) 6= i, and that σ fixes i if σ(i) = i. There are many interesting subgroups of Sn. For example, the subset Hn defined by Hn = {σ ∈ Sn : σ(n) = n}