8.1: Additional Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
34. Prove that a finite group G having just one maximal subgroup must be a cyclic p-group, p prime.
35. Let a and b be two elements of S76. If a and b both have order 146 and ab=ba, what are the possible orders of the product ab?
37. Suppose that the group G is generated by a set X.
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Show that if gxg−1∈X for all x∈X, g∈G, then the commutator subgroup of G is generated by the set of all elements xyx−1y−1 for x,y∈X.
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Show that if x2=1 for all x∈X, then the subgroup H of G generated by the set of all elements xy for x,y∈X has index 1 or 2.
38. Suppose p≥3 and 2p−1 are both prime numbers (e.g., p=3,7,19,31,…). Prove, or disprove by example, that every group of order p(2p−1) is commutative.
39. Let H be a subgroup of a group G. Prove or disprove the following:
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If G is finite and P is a Sylow p-subgroup, then H∩P is a Sylow p-subgroup of H.
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If G is finite, P is a Sylow p-subgroup, and H⊃NG(P), then NG(H)=H.
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If g is an element of G such that gHg−1⊂H, then g∈NG(H).
40. Prove that there is no simple group of order 616.
41. Let n and k be integers 1≤k≤n. Let H be the subgroup of Sn generated by the cycle (a1…ak). Find the order of the centralizer of H in Sn. Then find the order of the normalizer of H in Sn. [The centralizer of H is the set of g∈G such ghg−1=h for all h∈H. It is again a subgroup of G.]
42. Prove or disprove the following statement: if H is a subgroup of an infinite group G, then for all x∈G, xHx−1⊂H⟹x−1Hx⊂H.
43. Let H be a finite normal subgroup of a group G, and let g be an element of G. Suppose that g has order n and that the only element of H that commutes with g is 1. Show that:
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the mapping h↦g−1h−1gh is a bijection from H to H;
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the coset gH consists of elements of G of order n.
44. Show that if a permutation in a subgroup G of Sn maps x to y, then the normalizers of the stabilizers \Stab(x) and \Stab(y) of x and y have the same order.
45. Prove that if all Sylow subgroups of a finite group G are normal and abelian, then the group is abelian.
46. A group is generated by two elements a and b satisfying the relations: a3=b2, am=1, bn=1, where m and n are positive integers. For what values of m and n can G be infinite.
47. Show that the group G generated by elements x and y with defining relations x2=y3=(xy)4=1 is a finite solvable group, and find the order of G and its successive derived subgroups G′, G′′, G′′′.
48. A group G is generated by a normal set X of elements of order 2. Show that the commutator subgroup G′ of G is generated by all squares of products xy of pairs of elements of X.
49. Determine the normalizer N in \GLn(F) of the subgroup H of diagonal matrices, and prove that N/H is isomorphic to the symmetric group Sn.
50. Let G be a group with generators x and y and defining relations x2, y5, (xy)4. What is the index in G of the commutator group G′ of G.
51. Let G be a finite group, and H the subgroup generated by the elements of odd order. Show that H is normal, and that the order of G/H is a power of 2.
52. Let G be a finite group, and P a Sylow p-subgroup. Show that if H is a subgroup of G such that NG(P)⊂H⊂G, then
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the normalizer of H in G is H;
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(G:H)≡1 (mod p).
53. Let G be a group of order 33⋅25. Show that G is solvable. (Hint: A first step is to find a normal subgroup of order 11 using the Sylow theorems.)
54. Suppose that α is an endomorphism of the group G that maps G onto G and commutes with all inner automorphisms of G. Show that if G is its own commutator subgroup, then αx=x for all x in G.
55. Let G be a finite group with generators s and t each of order 2. Let n=(G:1)/2.
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Show that G has a cyclic subgroup of order n. Now assume n odd.
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Describe all conjugacy classes of G.
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Describe all subgroups of G of the form C(x)={y∈G|xy=yx}, x∈G.
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Describe all cyclic subgroups of G.
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Describe all subgroups of G in terms of (b) and (d).
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Verify that any two p-subgroups of G are conjugate (p prime).
56. Let G act transitively on a set X. Let N be a normal subgroup of G, and let Y be the set of orbits of N in X. Prove that:
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There is a natural action of G on Y which is transitive and shows that every orbit of N on X has the same cardinality.
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Show by example that if N is not normal then its orbits need not have the same cardinality.
57. Prove that every maximal subgroup of a finite p-group is normal of prime index (p is prime).
58. A group G is metacyclic if it has a cyclic normal subgroup N with cyclic quotient G/N. Prove that subgroups and quotient groups of metacyclic groups are metacyclic. Prove or disprove that direct products of metacyclic groups are metacylic.
59. Let G be a group acting doubly transitively on X, and let x∈X. Prove that:
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The stabilizer Gx of x is a maximal subgroup of G.
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If N is a normal subgroup of G, then either N is contained in Gx or it acts transitively on X.
60. Let x,y be elements of a group G such that xyx−1=y5, x has order 3, and y≠1 has odd order. Find (with proof) the order of y.
61. Let H be a maximal subgroup of G, and let A be a normal subgroup of H and such that the conjugates of A in G generate it.
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Prove that if N is a normal subgroup of G, then either N⊂H or G=NA.
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Let M be the intersection of the conjugates of H in G. Prove that if G is equal to its commutator subgroup and A is abelian, then G/M is a simple group.
62. (a) Prove that the centre of a nonabelian group of order p3, p prime, has order p.
(b) Exhibit a nonabelian group of order 16 whose centre is not cyclic.
63. Show that the group with generators α and β and defining relations
α2=β2=(αβ)3=1
is isomorphic with the symmetric group S3 of degree 3 by giving, with proof, an explicit isomorphism.
64. Prove or give a counter-example:
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Every group of order 30 has a normal subgroup of order 15.
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Every group of order 30 is nilpotent.
65. Let t∈Z, and let G be the group with generators x,y and relations xyx−1=yt, x3=1.
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Find necessary and sufficient conditions on t for G to be finite.
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In case G is finite, determine its order.
66. Let G be a group of order pq, p≠q primes.
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Prove G is solvable.
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Prove that G is nilpotent ⟺ G is abelian ⟺ G is cyclic.
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Is G always nilpotent? (Prove or find a counterexample.)
67. Let X be a set with pn elements, p prime, and let G be a finite group acting transitively on X. Prove that every Sylow p-subgroup of G acts transitively on X.
68. Let G=⟨a,b,c∣bc=cb, a4 aca−1=c, aba−1=bc⟩. Determine the order of G and find the derived series of G.
69. Let N be a nontrivial normal subgroup of a nilpotent group G. Prove that N∩Z(G)≠1.
70. Do not assume Sylow’s theorems in this problem.
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Let H be a subgroup of a finite group G, and P a Sylow p-subgroup of G. Prove that there exists an x∈G such that xPx−1∩H is a Sylow p-subgroup of H.
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Prove that the group of n×n matrices is a Sylow p-subgroup of \GLn(Fp).
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Indicate how (a) and (b) can be used to prove that any finite group has a Sylow p-subgroup.
71. Suppose H is a normal subgroup of a finite group G such that G/H is cyclic of order n, where n is relatively prime to (G:1). Prove that G is equal to the semidirect product H⋊S with S a cyclic subgroup of G of order n.
72. Let H be a minimal normal subgroup of a finite solvable group G. Prove that H is isomorphic to a direct sum of cyclic groups of order p for some prime p.
73. (a) Prove that subgroups A and B of a group G are of finite index in G if and only if A∩B is of finite index in G.
(b) An element x of a group G is said to be an FC-element if its centralizer CG(x) has finite index in G. Prove that the set of all FC elements in G is a normal.
74. Let G be a group of order p2q2 for primes p>q. Prove that G has a normal subgroup of order pn for some n≥1.
75. (a) Let K be a finite nilpotent group, and let L be a subgroup of K such that L⋅δK=K, where δK is the derived subgroup. Prove that L=K. [You may assume that a finite group is nilpotent if and only if every maximal subgroup is normal.]
(b) Let G be a finite group. If G has a subgroup H such that both G/δH and H are nilpotent, prove that G is nilpotent.
76. Let G be a finite noncyclic p-group. Prove that the following are equivalent:
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(G:Z(G))≤p2.
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Every maximal subgroup of G is abelian.
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There exist at least two maximal subgroups that are abelian.
77. Prove that every group G of order 56 can be written (nontrivially) as a semidirect product. Find (with proofs) two non-isomorphic non-abelian groups of order 56.
78. Let G be a finite group and φ:G→G a homomorphism.
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Prove that there is an integer n≥0 such that φn(G)=φm(G) for all integers m≥n. Let α=φn.
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Prove that G is the semi-direct product of the subgroups \Kerα and \imα.
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Prove that \imα is normal in G or give a counterexample.
79. Let S be a set of representatives for the conjugacy classes in a finite group G and let H be a subgroup of G. Show that S⊂H⟹H=G.
80. Let G be a finite group.
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Prove that there is a unique normal subgroup K of G such that (i) G/K is solvable and (ii) if N is a normal subgroup and G/N is solvable, then N⊃K.
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Show that K is characteristic.
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Prove that K=[K,K] and that K=1 or K is nonsolvable.