Abstract Algebra Theory and Applications - Computer Science ...

40 CHAPTER 2 GROUPSTable 2.2. Symmetries of an equilateral triangle◦ id ρ 1 ρ 2 µ 1 µ 2 µ 3id id ρ 1 ρ 2 µ 1 µ 2 µ 3ρ 1 ρ 1 ρ 2 id µ 3 µ 1 µ 2ρ 2 ρ 2 id ρ 1 µ 2 µ 3 µ 1µ 1 µ 1 µ 2 µ 3 id ρ 1 ρ 2µ 2 µ 2 µ 3 µ 1 ρ 2 id ρ 1µ 3 µ 3 µ 1 µ 2 ρ 1 ρ 2 idleft to right, we compose functions right to left. We have(µ 1 ρ 1 )(A) = µ 1 (ρ 1 (A)) = µ 1 (B) = C(µ 1 ρ 1 )(B) = µ 1 (ρ 1 (B)) = µ 1 (C) = B(µ 1 ρ 1 )(C) = µ 1 (ρ 1 (C)) = µ 1 (A) = A.This is the same symmetry as µ 2 . Suppose we do these motions in theopposite order, ρ 1 then µ 1 . It is easy to determine that this is the sameas the symmetry µ 3 ; hence, ρ 1 µ 1 ≠ µ 1 ρ 1 . A multiplication table for thesymmetries of an equilateral triangle △ABC is given in Table 2.1.Notice that in the multiplication table for the symmetries of an equilateraltriangle, for every motion of the triangle α there is another motion α ′such that αα ′ = id; that is, for every motion there is another motion thattakes the triangle back to its original orientation.2.2 Definitions and ExamplesThe integers mod n and the symmetries of a triangle or a rectangle are bothexamples of groups. A binary operation or law of composition on a setG is a function G × G → G that assigns to each pair (a, b) ∈ G a uniqueelement a ◦ b, or ab in G, called the composition of a and b. A group (G, ◦)is a set G together with a law of composition (a, b) ↦→ a ◦ b that satisfies thefollowing axioms.• The law of composition is associative. That is,for a, b, c ∈ G.(a ◦ b) ◦ c = a ◦ (b ◦ c)