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.The order of a finite group is the number of elements that it con-tains.If G is a group containing n elements, we write |G| = n.The groupZ5 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 e0 are both identities in G.Then eg = ge = gand e0g = ge0 = g for all g ∈ G.We need to show that e = e0.If we thinkof e as the identity, then ee0 = e0; but if e0 is the identity, then ee0 = e.Combining these two equations, we have e = ee0 = e0.Inverses in a group are also unique.If g0 and g00 are both inverses of anelement g in a group G, then gg0 = g0g = e and gg00 = g00g = e.We wantto show that g0 = g00, but g0 = g0e = g0(gg00) = (g0g)g00 = eg00 = g00.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−1a−1.Proof.Let a, b ∈ G.Then abb−1a−1 = aea−1 = aa−1 = e.Similarly,b−1a−1ab = e.But by the previous proposition, inverses are unique; hence,(ab)−1 = b−1a−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.2.2DEFINITIONS AND EXAMPLES45It makes sense to write equations with group elements and group opera-tions.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.Multi-plying both sides of ax = b by a−1, we have x = ex = a−1ax = a−1b.To show uniqueness, suppose that x1 and x2 are both solutions of ax = b;then ax1 = b = ax2.So x1 = a−1ax1 = a−1ax2 = x2.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 g0 = e.For n ∈ N, wedefinegn = g · g · · · g|{z}n timesandg−n = g−1 · g−1 · · · g−1.|{z}n timesTheorem 2.8 In a group, the usual laws of exponents hold; that is, for allg, h ∈ G,1.gmgn = gm+n for all m, n ∈ Z;2.(gm)n = gmn for all m, n ∈ Z;3.(gh)n = (h−1g−1)−n for all n ∈ Z.Furthermore, if G is abelian, then(gh)n = gnhn.We will leave the proof of this theorem as an exercise.Notice that(gh)n 6= gnhn in general, since the group may not be abelian.If the groupis Z or Zn, we write the group operation additively and the exponentialoperation multiplicatively; that is, we write ng instead of gn.The laws ofexponents now become46CHAPTER 2GROUPS1.mg + ng = (m + n)g for all m, n ∈ Z;2.m(ng) = (mn)g for all m, n ∈ Z;3.m(g + h) = mg + mh for all n ∈ Z.It is important to realize that the last statement can be made only becauseZ and Zn are commutative groups.Historical NoteAlthough the first clear axiomatic definition of a group was not given until thelate 1800s, group-theoretic methods had been employed before this time in thedevelopment of many areas of mathematics, including geometry and the theory ofalgebraic equations.Joseph-Louis Lagrange used group-theoretic methods in a 1770–1771 memoir tostudy methods of solving polynomial equations.Later, Évariste Galois (1811–1832)succeeded in developing the mathematics necessary to determine exactly whichpolynomial equations could be solved in terms of the polynomials’ coefficients.Galois’ primary tool was group theory.The study of geometry was revolutionized in 1872 when Felix Klein proposedthat geometric spaces should be studied by examining those properties that areinvariant under a transformation of the space.Sophus Lie, a contemporary ofKlein, used group theory to study solutions of partial differential equations.One ofthe first modern treatments of group theory appeared in William Burnside’s TheTheory of Groups of Finite Order [1], first published in 1897.2.3SubgroupsDefinitions and ExamplesSometimes we wish to investigate smaller groups sitting inside a larger group.The set of even integers 2Z = {., −2, 0, 2, 4,.} is a group under theoperation of addition.This smaller group sits naturally inside of the groupof integers under addition.We define a subgroup H of a group G to be asubset H of G such that when the group operation of G is restricted to H,H is a group in its own right.Observe that every group G with at least twoelements will always have at least two subgroups, the subgroup consisting ofthe identity element alone and the entire group itself.The subgroup H = {e}of a group G is called the trivial subgroup.A subgroup that is a propersubset of G is called a proper subgroup.In many of the examples that we2.3SUBGROUPS47have investigated up to this point, there exist other subgroups besides thetrivial and improper subgroups.Example 10.Consider the set of nonzero real numbers,∗R , with the groupoperation of multiplication.The identity of this group is 1 and the inverseof any element a ∈∗R is just 1/a [ Pobierz całość w formacie PDF ]
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.The order of a finite group is the number of elements that it con-tains.If G is a group containing n elements, we write |G| = n.The groupZ5 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 e0 are both identities in G.Then eg = ge = gand e0g = ge0 = g for all g ∈ G.We need to show that e = e0.If we thinkof e as the identity, then ee0 = e0; but if e0 is the identity, then ee0 = e.Combining these two equations, we have e = ee0 = e0.Inverses in a group are also unique.If g0 and g00 are both inverses of anelement g in a group G, then gg0 = g0g = e and gg00 = g00g = e.We wantto show that g0 = g00, but g0 = g0e = g0(gg00) = (g0g)g00 = eg00 = g00.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−1a−1.Proof.Let a, b ∈ G.Then abb−1a−1 = aea−1 = aa−1 = e.Similarly,b−1a−1ab = e.But by the previous proposition, inverses are unique; hence,(ab)−1 = b−1a−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.2.2DEFINITIONS AND EXAMPLES45It makes sense to write equations with group elements and group opera-tions.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.Multi-plying both sides of ax = b by a−1, we have x = ex = a−1ax = a−1b.To show uniqueness, suppose that x1 and x2 are both solutions of ax = b;then ax1 = b = ax2.So x1 = a−1ax1 = a−1ax2 = x2.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 g0 = e.For n ∈ N, wedefinegn = g · g · · · g|{z}n timesandg−n = g−1 · g−1 · · · g−1.|{z}n timesTheorem 2.8 In a group, the usual laws of exponents hold; that is, for allg, h ∈ G,1.gmgn = gm+n for all m, n ∈ Z;2.(gm)n = gmn for all m, n ∈ Z;3.(gh)n = (h−1g−1)−n for all n ∈ Z.Furthermore, if G is abelian, then(gh)n = gnhn.We will leave the proof of this theorem as an exercise.Notice that(gh)n 6= gnhn in general, since the group may not be abelian.If the groupis Z or Zn, we write the group operation additively and the exponentialoperation multiplicatively; that is, we write ng instead of gn.The laws ofexponents now become46CHAPTER 2GROUPS1.mg + ng = (m + n)g for all m, n ∈ Z;2.m(ng) = (mn)g for all m, n ∈ Z;3.m(g + h) = mg + mh for all n ∈ Z.It is important to realize that the last statement can be made only becauseZ and Zn are commutative groups.Historical NoteAlthough the first clear axiomatic definition of a group was not given until thelate 1800s, group-theoretic methods had been employed before this time in thedevelopment of many areas of mathematics, including geometry and the theory ofalgebraic equations.Joseph-Louis Lagrange used group-theoretic methods in a 1770–1771 memoir tostudy methods of solving polynomial equations.Later, Évariste Galois (1811–1832)succeeded in developing the mathematics necessary to determine exactly whichpolynomial equations could be solved in terms of the polynomials’ coefficients.Galois’ primary tool was group theory.The study of geometry was revolutionized in 1872 when Felix Klein proposedthat geometric spaces should be studied by examining those properties that areinvariant under a transformation of the space.Sophus Lie, a contemporary ofKlein, used group theory to study solutions of partial differential equations.One ofthe first modern treatments of group theory appeared in William Burnside’s TheTheory of Groups of Finite Order [1], first published in 1897.2.3SubgroupsDefinitions and ExamplesSometimes we wish to investigate smaller groups sitting inside a larger group.The set of even integers 2Z = {., −2, 0, 2, 4,.} is a group under theoperation of addition.This smaller group sits naturally inside of the groupof integers under addition.We define a subgroup H of a group G to be asubset H of G such that when the group operation of G is restricted to H,H is a group in its own right.Observe that every group G with at least twoelements will always have at least two subgroups, the subgroup consisting ofthe identity element alone and the entire group itself.The subgroup H = {e}of a group G is called the trivial subgroup.A subgroup that is a propersubset of G is called a proper subgroup.In many of the examples that we2.3SUBGROUPS47have investigated up to this point, there exist other subgroups besides thetrivial and improper subgroups.Example 10.Consider the set of nonzero real numbers,∗R , with the groupoperation of multiplication.The identity of this group is 1 and the inverseof any element a ∈∗R is just 1/a [ Pobierz całość w formacie PDF ]