204 Groups 69. In D 55. In D, let K Ro, D, D', R80. Show that K {Ro, D} and let L 4 L
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204 Groups 69. In D 55. In D, let K Ro, D, D', R80. Show that K {Ro, D} and let L 4 L<A D, but that K is not normal in D4. (Normality is not transitive Compare Exercise 4, Supplementary Exercises for Chapters 5-8 con 56. Show that the intersection of two normal subgroups of G is a nor- 70. Pro 71. If 72. If tair mal subgroup of G. Generalize. 57. Give an example of subgroups H and K of a group G such that HK is not a subgroup of G. 58. If N and M are normal subgroups of G, prove that NM is also a normal subgroup of G. 59. Let N be a normal subgroup of a group G. If N is cyclic, prove that every subgroup of N is also normal in G. (This exercise is referred to in Chapter 24.) 60. Without looking at inner automorphisms of D, determine the num- ber of such automorphisms. 61. Let H be a normal subgroup of a finite group G and let x E G. If ged(lxl, IGIHI) in Chapter 25.) 62. Let G be a group and let G' be the subgroup of G generated by the 73. Pr 74. Le inc an 75. Le a s ele 76. Su 1, show that x E H. (This exercise is referred to or 77. Le Sh (xly xy I x, y E G. (See Exercise 3, Supplementary set S 78. A Exercises for Chapters 5-8, for a more complete description of G'.) SU a. Prove that G' is normal in G. prг b. Prove that G/G' is Abelian. m c. If GIN is Abelian, prove that G'<N. d. Prove that if H is a subgroup of G and G' <H, then H is normal 79. Le or in G. 63. If N is a normal subgroup of G and IGINI m, show that xm E N 1 for all x in G. Suggested F 64. Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup Micha nitely (2000 of G. 65. If G is non-Abelian, show that Aut(G) is not cyclic. The 66. Let IGI p"m, where p is prime and gcd(p, m) = 1. Suppose that H is a normal subgroup of G of order p". If K is a subgroup of G of order p, show that K C H. 67. Suppose that H is a normal subgroup of a finite group G. If GlH has an element of order n, show that G has an element of order n. Show, by example, that the assumption that G is finite is necessary 68. Recall that a subgroup N of a group G is called characteristic if (N) = N for all automorphisms of G. If is a characteristic subgroup of G, show that N is a normal subgroup of G. pro tex J. A.C Zn" F Th rel:
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