is a cycle of length three, since h (1) = 4, h (4) = 3 and h (3) = 1, leaving 2 and 5 untouched. We denote such a cycle by (1 4 3), but it could equally well be written (4 3 1) or (3 1 4) by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are disjoint if they have disjoint subsets of elements. Disjoint cycles commute : for example, in S 6 there is the equality (4 1 3)(2 5 6) = (2 5 6)(4 1 3). Every element of S n can be written as a product of disjoint cycles; this representation is unique up to the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.