Abstract Algebra Theory and Applications - Computer Science ...

2.2 DEFINITIONS AND EXAMPLES 45It makes sense to write equations with group elements and group operations.If a and b are two elements in a group G, does there exist an elementx ∈ G such that ax = b? If such an x does exist, is it unique? The followingproposition answers both of these questions positively.Proposition 2.6 Let G be a group and a and b be any two elements in G.Then the equations ax = b and xa = b have unique solutions in G.Proof. Suppose that ax = b. We must show that such an x exists. Multiplyingboth sides of ax = b by a −1 , we have x = ex = a −1 ax = a −1 b.To show uniqueness, suppose that x 1 and x 2 are both solutions of ax = b;then ax 1 = b = ax 2 . So x 1 = a −1 ax 1 = a −1 ax 2 = x 2 . The proof for theexistence and uniqueness of the solution of xa = b is similar.□Proposition 2.7 If G is a group and a, b, c ∈ G, then ba = ca implies b = cand ab = ac implies b = c.This proposition tells us that the right and left cancellation lawsare true in groups. We leave the proof as an exercise.We can use exponential notation for groups just as we do in ordinaryalgebra. If G is a group and g ∈ G, then we define g 0 = e. For n ∈ N, wedefineg n = g · g · · · g} {{ }n timesandg −n = g −1 · g −1 · · · g −1 .} {{ }n timesTheorem 2.8 In a group, the usual laws of exponents hold; that is, for allg, h ∈ G,1. g m g n = g m+n for all m, n ∈ Z;2. (g m ) n = g mn for all m, n ∈ Z;3. (gh) n = (h −1 g −1 ) −n for all n ∈ Z. Furthermore, if G is abelian, then(gh) n = g n h n .We will leave the proof of this theorem as an exercise. Notice that(gh) n ≠ g n h n in general, since the group may not be abelian. If the groupis Z or Z n , we write the group operation additively and the exponentialoperation multiplicatively; that is, we write ng instead of g n . The laws ofexponents now become