Remarks

• The cycle notation of a permutation g does not tell us in which S_n we are working in. This is in contrast to the matrix notation. So (1,2) might belong to S₂ just as well as to S₃. This yields no real confusion because of the natural identification of S_{n - 1} with the part of S_n consisting of all permutations fixing n:
  S_{n - 1} = {g S_n | n fix(g)}.
  

• The composition of permutations in S_n (where n > 2) is not commutative. This means that the products gh and hg are not always the same. If gh = hg, then we say that g and h commute.

  Two cycles c and c' are called disjoint if the intersection of their supports is empty. Two disjoint cycles always commute. (Prove this!)

  A cycle (a₁ , ..., a_m) also commutes with its inverse (a_m, ..., a₁).