6 Chapter 1. Groups I Notice that composition in S3 is not commutative it is easy to find permuta- tions α, β of {1, 2, 3} with αβ = βα. We now introduce some special permutations. Definition. Let i1,i2,...,ir be distinct integers in X = {1, 2,...,n}. If α ∈ Sn fixes4 the other integers in X (if any) and if α(i1) = i2, α(i2) = i3,...,α(ir−1) = ir, α(ir) = i1, then α is called an r-cycle. We also say that α is a cycle of length r, and we denote it by α = (i1 i2 . . . ir). The term cycle comes from the Greek word for circle. The cycle α=(i1 i2 . . . ir) can be pictured as a clockwise rotation of the circle, as in Figure 1.2. i i i i1 r 2 3 . . . . . . Figure 1.2. Cycle α = (i1 i2 . . . ir). A 2-cycle interchanges i1 and i2 and fixes everything else 2-cycles are also called transpositions. A 1-cycle is the identity, for it fixes every i thus, all 1- cycles are equal. We extend the cycle notation to 1-cycles, writing (i) = (1) for all i [after all, (i) sends i into i and fixes everything else]. There are r different cycle notations for any r-cycle α, since any ij can be taken as its “starting point”: α = (i1 i2 . . . ir) = (i2 i3 . . . ir i1) = · · · = (ir i1 i2 . . . ir−1). Definition. Two permutations α, β ∈ Sn are disjoint if every i moved by one is fixed by the other: if α(i) = i, then β(i) = i, and if β(j) = j, then α(j) = j. A family β1,...,βt of permutations is disjoint if each pair of them is disjoint. For example, two cycles (i1 . . . ir) and (j1 . . . js) are disjoint if and only if {i1,...,ir} ∩ {j1,...,js} = ∅. 4Let f : X → X be a function. If x ∈ X, then f fixes x if f(x) = x, and f moves x if f(x) = x.

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