Abstract Algebra Theory and Applications - Computer Science ...

74 CHAPTER 4 PERMUTATION GROUPSthis inconsistency. We will adopt the convention of multiplying permutationsright to left. To compute στ, do τ first and then σ. That is, by στ(x) wemean σ(τ(x)). (Another way of solving this problem would be to writefunctions on the right; that is, instead of writing σ(x), we could write (x)σ.We could also multiply permutations left to right to agree with the usualway of multiplying elements in a group. Certainly all of these methods havebeen used.Example 2. Permutation multiplication is not usually commutative. Letσ =τ =( )1 2 3 44 1 2 3( )1 2 3 4.2 1 4 3Thenbutστ =τσ =( 1 2 3 41 4 3 2( 1 2 3 43 2 1 4),).Cycle NotationThe notation that we have used to represent permutations up to this point iscumbersome, to say the least. To work effectively with permutation groups,we need a more streamlined method of writing down and manipulating permutations.A permutation σ ∈ S X is a cycle of length k if there exist elementsa 1 , a 2 , . . . , a k ∈ X such thatσ(a 1 ) = a 2σ(a 2 ) = a 3σ(a k ) = a 1and σ(x) = x for all other elements x ∈ X. We will write (a 1 , a 2 , . . . , a k ) todenote the cycle σ. Cycles are the building blocks of all permutations..