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

Extracted text: 204 Groups 69. In D 55. In D, let K Ro, D, D', R80. Show that K {Ro, D} and let L 4 L
Jun 04, 2022
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