Abstract Algebra Theory and Applications - Computer Science ...

44 CHAPTER 2 GROUPSis the inverse of z. It is easy to see that the remaining group axioms hold.A group is finite, or has finite order, if it contains a finite number ofelements; otherwise, the group is said to be infinite or to have infiniteorder. The order of a finite group is the number of elements that it contains.If G is a group containing n elements, we write |G| = n. The groupZ 5 is a finite group of order 5; the integers Z form an infinite group underaddition, and we sometimes write |Z| = ∞.Basic Properties of GroupsProposition 2.2 The identity element in a group G is unique; that is, thereexists only one element e ∈ G such that eg = ge = g for all g ∈ G.Proof. Suppose that e and e ′ are both identities in G. Then eg = ge = gand e ′ g = ge ′ = g for all g ∈ G. We need to show that e = e ′ . If we thinkof e as the identity, then ee ′ = e ′ ; but if e ′ is the identity, then ee ′ = e.Combining these two equations, we have e = ee ′ = e ′ .□Inverses in a group are also unique. If g ′ and g ′′ are both inverses of anelement g in a group G, then gg ′ = g ′ g = e and gg ′′ = g ′′ g = e. We wantto show that g ′ = g ′′ , but g ′ = g ′ e = g ′ (gg ′′ ) = (g ′ g)g ′′ = eg ′′ = g ′′ . Wesummarize this fact in the following proposition.Proposition 2.3 If g is any element in a group G, then the inverse of g,g −1 , is unique.Proposition 2.4 Let G be a group. If a, b ∈ G, then (ab) −1 = b −1 a −1 .Proof. Let a, b ∈ G. Then abb −1 a −1 = aea −1 = aa −1 = e. Similarly,b −1 a −1 ab = e. But by the previous proposition, inverses are unique; hence,(ab) −1 = b −1 a −1 .□Proposition 2.5 Let G be a group. For any a ∈ G, (a −1 ) −1 = a.Proof. Observe that a −1 (a −1 ) −1 = e. Consequently, multiplying bothsides of this equation by a, we have(a −1 ) −1 = e(a −1 ) −1 = aa −1 (a −1 ) −1 = ae = a.□