Abstract Algebra Theory and Applications - Computer Science ...

4.1 DEFINITIONS AND NOTATION 77disjoint sets X 3 , X 4 , . . .. Since X is a finite set, we are guaranteed that thisprocess will end and there will be only a finite number of these sets, say r.If σ i is the cycle defined by{ σ(x) x ∈ Xiσ i (x) =x x /∈ X i ,then σ = σ 1 σ 2 · · · σ r . Since the sets X 1 , X 2 , . . . , X r are disjoint, the cyclesσ 1 , σ 2 , . . . , σ r must also be disjoint.□Example 6. Letσ =τ =( )1 2 3 4 5 66 4 3 1 5 2( )1 2 3 4 5 6.3 2 1 5 6 4Using cycle notation, we can writeσ = (1624)τ = (13)(456)στ = (136)(245)τσ = (143)(256).Remark. From this point forward we will find it convenient to use cyclenotation to represent permutations. When using cycle notation, we oftendenote the identity permutation by (1).TranspositionsThe simplest permutation is a cycle of length 2.transpositions. SinceSuch cycles are called(a 1 , a 2 , . . . , a n ) = (a 1 a n )(a 1 a n−1 ) · · · (a 1 a 3 )(a 1 a 2 ),any cycle can be written as the product of transpositions, leading to thefollowing proposition.Proposition 4.4 Any permutation of a finite set containing at least twoelements can be written as the product of transpositions.