Abstract Algebra Theory and Applications - Computer Science ...

4Permutation GroupsPermutation groups are central to the study of geometric symmetries and toGalois theory, the study of finding solutions of polynomial equations. Theyalso provide abundant examples of nonabelian groups.Let us recall for a moment the symmetries of the equilateral triangle△ABC from Chapter 2. The symmetries actually consist of permutationsof the three vertices, where a permutation of the set S = {A, B, C} is aone-to-one and onto map π : S → S. The three vertices have the followingsix permutations.( A B CA B C( A B CA C B) ( A B CC A B) ( A B CC B A) ( A B CB C A) ( A B CB A C))We have used the array( A B CB C A)to denote the permutation that sends A to B, B to C, and C to A. That is,A ↦→ BB ↦→ CC ↦→ A.The symmetries of a triangle form a group. In this chapter we will studygroups of this type.72