Abel's theorem in problems and solutions - School of Mathematics

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138 Problems of Chapter 1 element of the group F. Since is a surjective homomorphism there exists in the group G an element such that Thus (see 134) and Since N is a normal subgroup of the group G the element belongs to and is thus a normal subgroup of the group F. 155. Let be an arbitrary commutator in F. Since is a surjective homomorphism there exist in the group G two elements and such that and Thus (see 134) and Since the element belongs to the commutator belongs to But since is a subgroup of F (see 150) and contains all commutators‚ contains the entire commutant On the other hand‚ is contained in (see 153). Hence The equality does not hold in general. For example‚ the mapping of the group into the group consisting of a sole element‚ such that and is a surjective homomorphism. One has and 156. Answer‚ a) Yes: the group is commutative; b) yes (see 118); c) yes (see 118); d) yes (see 118); e) yes (see 120); f) yes (see 122 and (e)); g) yes (see 118 and (e)). 157. A given face can be sent onto any one of the 12 faces in 5 distinct ways. Answer. 60. 158. Answer. 1) 1; 2) 24; 3) 20; 4) 15. 159. Let and be two axes of the same type‚ (i.e.‚ two axes either through the centres of opposite faces‚ or through opposite vertices‚ or through the centres of opposite edges). There exists a rotation of the dodecahedron which sends the axis to the axis If is any (nontrivial) rotation about then the rotation sends (verify) the axis to itself‚ without reversing it. The element is not the identity‚ otherwise should be the identical rotation about the axis Hence if a normal subgroup in the group of rotations contains one rotation about one axis then it contains at least one rotation about every one of the axes of the same type. A subgroup of rotations of the dodecahedron about one axis have an order (according to the type of axis) equal to 5‚ 3‚ or 2. Since 5‚ 3‚ and 2 are prime numbers‚ any element (different from is a generator of this subgroup (see 34)‚ i.e.‚ it generates the entire
Solutions 139 subgroup. Hence if a normal subgroup of the group of rotations of the dodecahedron contains one rotation about some axis then it contains all the rotations about all the axes of that type. 160. From the result of Problem 159 it follows that a normal subgroup in the group of the rotations of the dodecahedron must consist in some one of the classes 1–4‚ necessarily including class 1. The order of the normal subgroup must be a divisor of the order of the group of the rotations of the dodecahedron (i.e.‚ 60) (see Lagrange’s theorem‚ §1.8). It is easy to verify (see 158) that this is possible only if the normal subgroup contains only class 1 or contains all the classes. 161. Since the group G is not commutative‚ the commutant Since K(G) is a normal subgroup of G (see 116) it follows from the hypothesis of the problem that K(G) = G. Therefore in the sequence G‚K(G)‚ … all groups coincide with G and this sequence cannot end with the unit group. Hence the group G is not soluble. 162. Let the group G be soluble. Thus there exists an integer such that the subgroup is the unit group. If H is a subgroup of the group G then K(H) is contained in K(G)‚ K(K(H)) is contained in K(K(G))‚ etc.. Since the subgroup is contained in and the subgroup is the unit‚ is also the unit subgroup. It follows that the subgroup H is soluble. 163. Denote by K(G) the commutant in the group G and by the subgroup Since is a surjective homomorphism of the group G onto F‚ (see 155)‚ and in general Since the group G is soluble‚ for a certain the subgroup will be the unit group. But and thus will be also the unit group. Hence the group F is soluble. 164. Answer. For example‚ where G is the group of the rotations of the tetrahedron. 165. Hint. Consider the natural homomorphism of the group G into the quotient group G/N (§1.13). Use later the result of Problem 163.
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ABEL’S THEOREM IN PROBLEMS AND SO
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Abel’s Theorem in Problems and So
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Contents Preface for the English ed
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A.10.3 The integrable case A.11 Top
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Preface to the English edition by V
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of the differential equations. Unfo
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Preface In high school algebraic eq
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Introduction We begin this book by
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Introduction 3 where by is indicate
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Introduction 5 By Viète’s theore
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Introduction 7 We will be able to p
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Chapter 1 Groups 1.1 Examples In ar
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Groups 11 the triangle ABC into its
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Groups 13 7. Find all symmetries of
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Groups 15 notations (see the soluti
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Groups 17 element, which we will de
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Groups 19 35. Suppose that is the m
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Groups 21 If a group is isomorphic
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Groups 23 tude); (rotation by 180°
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Groups 25 Hence left cosets generat
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Groups 27 as a permutation of the t
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Groups 29 102. Let be the order of
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Groups 31 111. Find all normal subg
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Groups 33 FIGURE 8 127. Prove that
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Groups 35 137. Prove that ker is a
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Groups 37 We now observe what happe
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Groups 39 FIGURE 12 vertices; 4) ro
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Groups 41 Every permutation of degr
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Groups 43 180. Prove that by multip
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Chapter 2 The complex numbers When
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The complex numbers 47 195. Prove t
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The complex numbers 49 where is the
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The complex numbers 51 Consequently
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The complex numbers 53 For complex
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The complex numbers 55 think that n
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The complex numbers 57 214. Let us
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The complex numbers 59 219. Prove t
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The complex numbers 61 functions (s
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The complex numbers 63 of continuit
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The complex numbers 65 erations of
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The complex numbers 67 a) b) c) d)
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The complex numbers 69 REMARK. If a
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The complex numbers 71 2.8 Images o
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The complex numbers 73 where all th
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The complex numbers 75 in such func
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The complex numbers 77 FIGURE 26 no
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The complex numbers 79 C from to th
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The complex numbers 81 DEFINITION.
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The complex numbers 83 FIGURE 32 cu
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The complex numbers 85 to infinity
Page 103 and 104: The complex numbers 87 property is
Page 105 and 106: The complex numbers 89 means will b
Page 107 and 108: The complex numbers 91 For example,
Page 109 and 110: The complex numbers 93 of the Riema
Page 111 and 112: The complex numbers 95 in Figure 38
Page 113 and 114: The complex numbers 97 permutation
Page 115 and 116: The complex numbers 99 2.13 Monodro
Page 117 and 118: The complex numbers 101 First we pr
Page 119 and 120: The complex numbers 103 and prove t
Page 121 and 122: Chapter 3 Hints, Solutions, and Ans
Page 123 and 124: Solutions 107 positive integer unde
Page 125 and 126: Solutions 109 on the left and by on
Page 127 and 128: Solutions 111 39. a) See Table 8; b
Page 129 and 130: Solutions 113 of the real numbers i
Page 131 and 132: Solutions 115 to all and therefore
Page 133 and 134: Solutions 117 is a subgroup of the
Page 135 and 136: Solutions 119 hypothesis. The group
Page 137 and 138: Solutions 121 The given group is th
Page 139 and 140: Solutions 123 of symmetries of the
Page 141 and 142: Solutions 125 i.e., there exists an
Page 143 and 144: Solutions 127 Answer. The normal su
Page 145 and 146: Solutions 129 In this way the quoti
Page 147 and 148: Solutions 131 sends every into itse
Page 149 and 150: Solutions 133 ement of the commutan
Page 151 and 152: Solutions 135 143. Let be the three
Page 153: Solutions 137 150. See 57. Suppose
Page 157 and 158: Solutions 141 is commutative. Since
Page 159 and 160: Solutions 143 180. Since the lower
Page 161 and 162: Solutions 145 FIGURE 43 190. The pe
Page 163 and 164: Solutions 147 permutations of type
Page 165 and 166: Solutions 149 is a field. 195. We h
Page 167 and 168: Solutions 151 necessarily that and
Page 169 and 170: Solutions 153 Let C be the field of
Page 171 and 172: Solutions 155 in as numerators and
Page 173 and 174: Solutions 157 c) d) e) where 226. W
Page 175 and 176: Solutions 159 function of real argu
Page 177 and 178: Solutions 161 i.e., Consider the re
Page 179 and 180: Solutions 163 Choose as the smalles
Page 181 and 182: Solutions 165 Since the function is
Page 183 and 184: Solutions 167 FIGURE 55 FIGURE 56 F
Page 185 and 186: Solutions 169 FIGURE 60 257. Suppos
Page 187 and 188: Solutions 171 265. 266. 267. If the
Page 189 and 190: Solutions 173 following way. If the
Page 191 and 192: Solutions 175 for all such that and
Page 193 and 194: Solutions 177 b) In order for to va
Page 195 and 196: Solutions 179 to the condition coin
Page 197 and 198: Solutions 181 are continuous functi
Page 199 and 200: Solutions 183 turn around the other
Page 201 and 202: Solutions 185 sought is shown in Fi
Page 203 and 204: Solutions 187 FIGURE 86 FIGURE 87 p
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Solutions 189 sum of multi-valued f
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Solutions 191 316. a) Let and be th
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Solutions 193 by the formal method,
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Solutions 195 The correct scheme is
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Solutions 197 branch of the functio
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Solutions FIGURE 110 FIGURE 111 329
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Solutions 201 It follows that and b
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Solutions 203 340. To every sheet o
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Solutions 205 Since by hypothesis t
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Solutions 207 moving along a curve
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Drawings of Riemann surfaces 209 Th
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FIGURE 119 211
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FIGURE 121 213
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FIGURE 123 215
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FIGURE 125 217
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FIGURE 127 219
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Appendix by A. Khovanskii: Solvabil
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Solvability of Equations 223 functi
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Solvability of Equations 225 Suppos
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Solvability of Equations 227 where
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Solvability of Equations 229 finds
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Solvability of Equations 231 quadra
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Solvability of Equations 233 More p
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Solvability of Equations 235 group.
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Solvability of Equations 237 Every
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Solvability of Equations 239 The gr
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Solvability of Equations 241 FIG. 1
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Solvability of Equations 243 are ob
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Solvability of Equations 245 corres
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Solvability of Equations 249 up to
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Solvability of Equations 251 fined
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Solvability of Equations 253 the qu
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Solvability of Equations 255 the no
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Solvability of Equations 257 THEORE
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Bibliography [1] [2] [3] [4] [5] [6
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[23] [24] [25] [26] [27] [28] 263 V
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Appendix (V.I. Arnold) The topologi
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Index Abel’s theorem, 6, 103 addi
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pack of sheets, 96 parametric equat
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