210 CHAPTER 4 Abstract Algebra
(i) Give an automorphismσwhich takes vertex 1 to vertex 2 by
completing the following:σ:
1 2 3 4 5 6
↓ ↓ ↓ ↓ ↓ ↓
2
Compute the inverse of this automorphism.(ii) Give an automorphismτ which maps vertex 1 to 3 and fixes
vertex 5 (that isτ(5) = 5).τ :
1 2 3 4 5 6
↓ ↓ ↓ ↓ ↓ ↓
3 5
Compute the inverse of this automorphism.4.2.2 Abstract algebra—the concept of a binary operation
Intuitively, abinary operationon a setSis a rule for “multiplying”
elements ofStogether to form new elements.