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8.1: Additional Exercises

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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.

  1. Show that if gxg1X for all xX, gG, then the commutator subgroup of G is generated by the set of all elements xyx1y1 for x,yX.

  2. Show that if x2=1 for all xX, then the subgroup H of G generated by the set of all elements xy for x,yX has index 1 or 2.

38. Suppose p3 and 2p1 are both prime numbers (e.g., p=3,7,19,31,). Prove, or disprove by example, that every group of order p(2p1) is commutative.

39. Let H be a subgroup of a group G. Prove or disprove the following:

  1. If G is finite and P is a Sylow p-subgroup, then HP is a Sylow p-subgroup of H.

  2. If G is finite, P is a Sylow p-subgroup, and HNG(P), then NG(H)=H.

  3. If g is an element of G such that gHg1H, then gNG(H).

40. Prove that there is no simple group of order 616.

41. Let n and k be integers 1kn. Let H be the subgroup of Sn generated by the cycle (a1ak). 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 gG such ghg1=h for all hH. 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 xG, xHx1Hx1HxH.

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:

  1. the mapping hg1h1gh is a bijection from H to H;

  2. 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)HG, then

  1. the normalizer of H in G is H;

  2. (G:H)1 (mod p).

53. Let G be a group of order 3325. 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.

  1. Show that G has a cyclic subgroup of order n. Now assume n odd.

  2. Describe all conjugacy classes of G.

  3. Describe all subgroups of G of the form C(x)={yG|xy=yx}, xG.

  4. Describe all cyclic subgroups of G.

  5. Describe all subgroups of G in terms of (b) and (d).

  6. 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:

  1. 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.

  2. 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 xX. Prove that:

  1. The stabilizer Gx of x is a maximal subgroup of G.

  2. 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 xyx1=y5, x has order 3, and y1 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.

  1. Prove that if N is a normal subgroup of G, then either NH or G=NA.

  2. 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:

  1. Every group of order 30 has a normal subgroup of order 15.

  2. Every group of order 30 is nilpotent.

65. Let tZ, and let G be the group with generators x,y and relations xyx1=yt, x3=1.

  1. Find necessary and sufficient conditions on t for G to be finite.

  2. In case G is finite, determine its order.

66. Let G be a group of order pq, pq primes.

  1. Prove G is solvable.

  2. Prove that G is nilpotent G is abelian G is cyclic.

  3. 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,cbc=cb, a4 aca1=c, aba1=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 NZ(G)1.

70. Do not assume Sylow’s theorems in this problem.

  1. Let H be a subgroup of a finite group G, and P a Sylow p-subgroup of G. Prove that there exists an xG such that xPx1H is a Sylow p-subgroup of H.

  2. Prove that the group of n×n matrices is a Sylow p-subgroup of \GLn(Fp).

  3. 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 HS 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 AB 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 n1.

75. (a) Let K be a finite nilpotent group, and let L be a subgroup of K such that LδK=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:

  1. (G:Z(G))p2.

  2. Every maximal subgroup of G is abelian.

  3. 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:GG a homomorphism.

  1. Prove that there is an integer n0 such that φn(G)=φm(G) for all integers mn. Let α=φn.

  2. Prove that G is the semi-direct product of the subgroups \Kerα and \imα.

  3. 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 SHH=G.

80. Let G be a finite group.

  1. 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 NK.

  2. Show that K is characteristic.

  3. Prove that K=[K,K] and that K=1 or K is nonsolvable.


This page titled 8.1: Additional Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James S. Milne.

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