8.1: Additional Exercises

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 \(S_{76}\). 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 \(gxg^{-1}\in X\) for all \(x\in X,\) \(g\in G\), then the commutator subgroup of \(G\) is generated by the set of all elements \(xyx^{-1}y^{-1}\) for \(x,y\in X\).

2. Show that if \(x^{2}=1\) for all \(x\in X\), then the subgroup \(H\) of \(G\) generated by the set of all elements \(xy\) for \(x,y\in X\) has index \(1\) or \(2\).

38. Suppose \(p\geq3\) and \(2p-1\) are both prime numbers (e.g., \(p=3,7,19,31,\ldots)\). 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:

1. If \(G\) is finite and \(P\) is a Sylow \(p\)-subgroup, then \(H\cap P\) is a Sylow \(p\)-subgroup of \(H\).

2. If \(G\) is finite, \(P\) is a Sylow \(p\)-subgroup, and \(H\supset N_{G}(P)\), then \(N_{G}(H)=H\).

3. If \(g\) is an element of \(G\) such that \(gHg^{-1}\subset H\), then \(g\in N_{G}(H)\).

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

41. Let \(n\) and \(k\) be integers \(1\leq k\leq n\). Let \(H\) be the subgroup of \(S_{n}\) generated by the cycle \((a_{1}\ldots a_{k})\). Find the order of the centralizer of \(H\) in \(S_{n}\). Then find the order of the normalizer of \(H\) in \(S_{n}\). [The centralizer of \(H\) is the set of \(g\in G\) such \(ghg^{-1}=h\) for all \(h\in H\). 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 \(x\in G\), \(xHx^{-1}\subset H\implies x^{-1}Hx\subset H\).

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:

1. the mapping \(h\mapsto g^{-1}h^{-1}gh\) 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 \(S_{n}\) 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: \(a^{3}=b^{2}\), \(a^{m}=1\), \(b^{n}=1\), 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 \(x^{2}=y^{3}=(xy)^{4}=1\) is a finite solvable group, and find the order of \(G\) and its successive derived subgroups \(G^{\prime}\), \(G^{\prime\prime}\), \(G^{\prime\prime\prime}\).

48. A group \(G\) is generated by a normal set \(X\) of elements of order \(2\). Show that the commutator subgroup \(G^{\prime}\) of \(G\) is generated by all squares of products \(xy\) of pairs of elements of \(X\).

49. Determine the normalizer \(N\) in \(\GL_{n}(F)\) of the subgroup \(H\) of diagonal matrices, and prove that \(N/H\) is isomorphic to the symmetric group \(S_{n}\).

50. Let \(G\) be a group with generators \(x\) and \(y\) and defining relations \(x^{2}\), \(y^{5}\), \((xy)^{4}\). What is the index in \(G\) of the commutator group \(G^{\prime}\) 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 \(N_{G}(P)\subset H\subset G\), then

1. the normalizer of \(H\) in \(G\) is \(H\);

2. \((G:H)\equiv1\) (mod \(p\)).

53. Let \(G\) be a group of order \(33\cdot25\). 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 \(\alpha\) 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 \(\alpha x=x\) 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 \(C(x)=\{y\in G|xy=yx\}\), \(x\in G\).

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 \(x\in X\). Prove that:

1. The stabilizer \(G_{x}\) of \(x\) is a maximal subgroup of \(G\).

2. If \(N\) is a normal subgroup of \(G\), then either \(N\) is contained in \(G_{x}\) or it acts transitively on \(X\).

60. Let \(x,y\) be elements of a group \(G\) such that \(xyx^{-1}=y^{5}\), \(x\) has order \(3\), and \(y\neq1\) 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 \(N\subset H\) 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 \(p^{3}\), \(p\) prime, has order \(p\).

(b) Exhibit a nonabelian group of order \(16\) whose centre is not cyclic.

63. Show that the group with generators \(\alpha\) and \(\beta\) and defining relations

\[\alpha^{2}=\beta^{2}=(\alpha\beta)^{3}=1 \nonumber \]

is isomorphic with the symmetric group \(S_{3}\) 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 \(t\in\mathbb{Z}\), and let \(G\) be the group with generators \(x,y\) and relations \(xyx^{-1}=y^{t}\), \(x^{3}=1\).

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 \(pq\), \(p\neq q\) primes.

1. Prove \(G\) is solvable.

2. Prove that \(G\) is nilpotent \(\iff\) \(G\) is abelian \(\iff\) G is cyclic.

3. Is \(G\) always nilpotent? (Prove or find a counterexample.)

67. Let \(X\) be a set with \(p^{n}\) 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 \(G=\langle a,b,c\mid bc=cb\), \(a^{4}% =b^{2}=c^{2}=1,\) \(aca^{-1}=c\), \(aba^{-1}=bc\rangle\). 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 \(N\cap Z(G)\neq1\).

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 \(x\in G\) such that \(xPx^{-1}\cap H\) is a Sylow \(p\)-subgroup of \(H\).

2. Prove that the group of \(n\times n\) matrices \(% \begin{pmatrix} 1 & \ast & \ldots\\ 0 & 1 & \cdots\\ & \ldots & \\ 0 & & 1 \end{pmatrix}\) is a Sylow \(p\)-subgroup of \(\GL_{n}(\mathbb{F}_{p})\).

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 \(H\rtimes S\) 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 \(A\cap B\) is of finite index in \(G\).

(b) An element \(x\) of a group \(G\) is said to be an FC-element if its centralizer \(C_{G}(x)\) 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 \(p^{2}q^{2}\) for primes \(p>q\). Prove that \(G\) has a normal subgroup of order \(p^{n}\) for some \(n\geq1\).

75. (a) Let \(K\) be a finite nilpotent group, and let \(L\) be a subgroup of \(K\) such that \(L\cdot\delta K=K\), where \(\delta K\) 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 \(G/\delta H\) 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. \((G:Z(G))\leq p^{2}\).

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 \(\varphi :G\rightarrow G\) a homomorphism.

1. Prove that there is an integer \(n\geq0\) such that \(\varphi ^{n}(G)=\varphi^{m}(G)\) for all integers \(m\geq n\). Let \(\alpha=\varphi^{n}\).

2. Prove that \(G\) is the semi-direct product of the subgroups \(\Ker\alpha\) and \(\im\alpha\).

3. Prove that \(\im\alpha\) 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 \(S\subset H\implies H=G\).

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 \(N\supset K\).

2. Show that \(K\) is characteristic.

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