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.EXERCISES199A group G is solvable if it has a composition series {Hi} such that allof the factor groups Hi+1/Hi are abelian.Solvable groups will play a fun-damental role when we study Galois theory and the solution of polynomialequations.Example 11.The group S4 is solvable sinceS4 ⊃ A4 ⊃ {(1), (12)(34), (13)(24), (14)(23)} ⊃ {(1)}has abelian factor groups; however, for n ≥ 5 the seriesSn ⊃ An ⊃ {(1)}is a composition series for Sn with a nonabelian factor group.Therefore, Snis not a solvable group for n ≥ 5.Exercises1.Find all of the abelian groups of order less than or equal to 40 up to isomor-phism.2.Find all of the abelian groups of order 200 up to isomorphism.3.Find all of the abelian groups of order 720 up to isomorphism.4.Find all of the composition series for each of the following groups.(a) Z12(b) Z48(c) The quaternions, Q8(d) D4(e) S3 × Z4(f ) S4(g) Sn, n ≥ 5(h) Q5.Show that the infinite direct product G = Z2 × Z2 × · · · is not finitelygenerated.6.Let G be an abelian group of order m.If n divides m, prove that G has asubgroup of order n.7.A group G is a torsion group if every element of G has finite order.Provethat a finitely generated torsion group must be finite.8.Let G, H, and K be finitely generated abelian groups.Show that if G × H ∼=G × K, then H ∼= K.Give a counterexample to show that this cannot betrue in general.9.Let G and H be solvable groups.Show that G × H is also solvable.200CHAPTER 11THE STRUCTURE OF GROUPS10.If G has a composition (principal) series and if N is a proper normal subgroupof G, show there exists a composition (principal) series containing N.11.Prove or disprove: Let N be a normal subgroup of G.If N and G/N havecomposition series, then G must also have a composition series.12.Let N be a normal subgroup of G.If N and G/N are solvable groups, showthat G is also a solvable group.13.Prove that G is a solvable group if and only if G has a series of subgroupsG = Pn ⊃ Pn−1 ⊃ · · · ⊃ P1 ⊃ P0 = {e}where Pi is normal in Pi+1 and the order of Pi+1/Pi is prime.14.Let G be a solvable group.Prove that any subgroup of G is also solvable.15.Let G be a solvable group and N a normal subgroup of G.Prove that G/Nis solvable.16.Prove that Dn is solvable for all integers n.17.Suppose that G has a composition series.If N is a normal subgroup of G,show that N and G/N also have composition series.18.Let G be a cyclic p-group with subgroups H and K.Prove that either H iscontained in K or K is contained in H.19.Suppose that G is a solvable group with order n ≥ 2.Show that G containsa normal nontrivial abelian subgroup.20.Recall that the commutator subgroup G0 of a group G is defined asthe subgroup of G generated by elements of the form a−1b−1ab for a, b ∈ G.We can define a series of subgroups of G by G(0) = G, G(1) = G0, andG(i+1) = (G(i))0.(a) Prove that G(i+1) is normal in (G(i))0.The series of subgroupsG(0) = G ⊃ G(1) ⊃ G(2) ⊃ · · ·is called the derived series of G.(b) Show that G is solvable if and only if G(n) = {e} for some integer n.21.Suppose that G is a solvable group with order n ≥ 2.Show that G containsa normal nontrivial abelian factor group.22.Zassenhaus Lemma.Let H and K be subgroups of a group G.Supposealso that H∗ and K∗ are normal subgroups of H and K respectively.Then(a) H∗(H ∩ K∗) is a normal subgroup of H∗(H ∩ K).(b) K∗(H∗ ∩ K) is a normal subgroup of K∗(H ∩ K).EXERCISES201(c)H∗(H ∩ K)/H∗(H ∩ K∗)∼=K∗(H ∩ K)/K∗(H∗ ∩ K)∼=(H ∩ K)/(H∗ ∩ K)(H ∩ K∗).[Hint: Use the diagram in Figure 11.1.The Zassenhaus Lemma is oftenreferred to as the Butterfly Lemma because of this diagram.]HKQQBQBQQBH ∩ KH∗(H ∩ K)QK∗(H ∩ K)BQQBBH∗(H ∩ K∗)K∗(H∗ ∩ K)BBQHA HQBHQAHBQHAHH∗QBHK∗QAHBQ@AQ@(H∗ ∩ K)(H ∩ K∗)QAQ@AQ@QAH ∩ K∗H∗ ∩ KFigure 11.1.The Zassenhaus Lemma23.Schreier’s Theorem.Use the Zassenhaus Lemma to prove that two sub-normal (normal) series of a group G have isomorphic refinements.24.Use Schreier’s Theorem to prove the Jordan-Hölder Theorem.Programming ExercisesWrite a program that will compute all possible abelian groups of order n.What isthe largest n for which your program will work?References and Suggested ReadingsEach of the following references contains a proof of the Fundamental Theorem ofFinitely Generated Abelian Groups.[1] Hungerford, T.W.Algebra.Springer-Verlag, New York, 1974.202CHAPTER 11THE STRUCTURE OF GROUPS[2] Lang, S.Algebra.3rd ed.Addison-Wesley, Reading, MA, 1992.[3] Rotman, J.J.An Introduction to the Theory of Groups.3rd ed.Allyn andBacon, Boston, 1984.12Group ActionsGroup actions generalize group multiplication.If G is a group and X is anarbitrary set, a group action of an element g ∈ G and x ∈ X is a product,gx, living in X.Many problems in algebra may best be attacked via groupactions.For example, the proofs of the Sylow theorems and of Burnside’sCounting Theorem are most easily understood when they are formulated interms of group actions.12.1Groups Acting on SetsLet X be a set and G be a group.A (left) action of G on X is a mapG × X → X given by (g, x) 7→ gx, where1.ex = x for all x ∈ X;2.(g1g2)x = g1(g2x) for all x ∈ X and all g1, g2 ∈ G.Under these considerations X is called a G-set.Notice that we are notrequiring X to be related to G in any way [ Pobierz całość w formacie PDF ]