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

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14 Chapter 1 where the first row contains all the elements of the given set, and the second row indicates all the corresponding images under the permutation. Since the transformation is one to one, for every transformation there exists the inverse transformation which is defined in the following way: if then So in Example 1 Therefore i.e., 10. Find the inverse transformations of all symmetries of the equilateral triangle (see Examples 1 and 2). 11. Consider the transformation of all real numbers given by Find the inverse transformation. The multiplication of the transformations and is defined as (the transformation is done first, afterwards). If and axe transformations of the set M then is also a transformation of set M. DEFINITION. Suppose that a set G of transformations possesses the following properties: 1) if two transformations and belong to G, then their product also belongs to G; 2) if a transformation belongs to G then its inverse transformation belongs to G. In this case we call such a set of transformations a group of transformations. It is not difficult to verify that the sets of transformations considered in Examples 1–6 are, in fact, groups of transformations. 12. Prove that any group of transformations contains the identical transformation such that for every element A of the set M. 13. Prove that for any transformation 14. Prove that for any three transformations and the following equality holds 3 : 1.3 Groups To solve Problems 6 and 7 we wrote the multiplication tables for the symmetries of the rhombus and of the rectangle. It has turned out that in our 3 This equality is true not only for transformations but also for any three mappings and such that
Groups 15 notations (see the solutions) these tables coincide. For many purposes it is natural to consider such groups of transformations as coinciding. Therefore we shall consider abstract objects rather than sets of real elements (in our case of transformations). Furthermore, we shall consider those binary operations on arbitrary sets which possess the basic properties of the binary operation in a group of transformations. Thus any binary operation will be called a multiplication; if to the pair there corresponds the element we call the product of and and we write In some special cases the binary operation will be called differently, for example, composition, addition, etc.. DEFINITION. A set G of elements of an arbitrary nature, on which one can define a binary operation such that the following conditions are satisfied, is called a group: 1) associativity : for any elements and of G; 2) in G there is an element such that for every element of G; such element is called the unit (or neutral element) of group G; 3) for every element of G there is in G an element such that such an element is called the inverse of element From the results of Problems 12–14 we see that any group of transformations is a group (in some sense the converse statement is also true (see 55)). In this way we have already seen a lot of examples of groups. All these groups contain a finite number of elements: such groups are called finite groups. The number of elements of a finite group is called the order of the group. Groups containing an infinite number of elements are called infinite groups. Let us give some examples of infinite groups. EXAMPLE 7. Consider the set of all integer numbers. In this set we shall take as binary operation the usual addition. We thus obtain a group. Indeed, the role of the unit element is played by 0, because for every integer Moreover, for every there exists the inverse element (which is called in this case the opposite element), because The associativity follows from the rules of arithmetic. The obtained group is called the group of integers under addition. 15. Consider the following sets: 1) all the real numbers; 2) all the real numbers without zero. Say whether the sets 1 and 2 form a group under multiplication.
Page 3 and 4: ABEL’S THEOREM IN PROBLEMS AND SO
Page 5 and 6: Abel’s Theorem in Problems and So
Page 7 and 8: Contents Preface for the English ed
Page 9 and 10: A.10.3 The integrable case A.11 Top
Page 11 and 12: Preface to the English edition by V
Page 13 and 14: of the differential equations. Unfo
Page 15 and 16: Preface In high school algebraic eq
Page 17 and 18: Introduction We begin this book by
Page 19 and 20: Introduction 3 where by is indicate
Page 21 and 22: Introduction 5 By Viète’s theore
Page 23 and 24: Introduction 7 We will be able to p
Page 25 and 26: Chapter 1 Groups 1.1 Examples In ar
Page 27 and 28: Groups 11 the triangle ABC into its
Page 29: Groups 13 7. Find all symmetries of
Page 33 and 34: Groups 17 element, which we will de
Page 35 and 36: Groups 19 35. Suppose that is the m
Page 37 and 38: Groups 21 If a group is isomorphic
Page 39 and 40: Groups 23 tude); (rotation by 180°
Page 41 and 42: Groups 25 Hence left cosets generat
Page 43 and 44: Groups 27 as a permutation of the t
Page 45 and 46: Groups 29 102. Let be the order of
Page 47 and 48: Groups 31 111. Find all normal subg
Page 49 and 50: Groups 33 FIGURE 8 127. Prove that
Page 51 and 52: Groups 35 137. Prove that ker is a
Page 53 and 54: Groups 37 We now observe what happe
Page 55 and 56: Groups 39 FIGURE 12 vertices; 4) ro
Page 57 and 58: Groups 41 Every permutation of degr
Page 59 and 60: Groups 43 180. Prove that by multip
Page 61 and 62: Chapter 2 The complex numbers When
Page 63 and 64: The complex numbers 47 195. Prove t
Page 65 and 66: The complex numbers 49 where is the
Page 67 and 68: The complex numbers 51 Consequently
Page 69 and 70: The complex numbers 53 For complex
Page 71 and 72: The complex numbers 55 think that n
Page 73 and 74: The complex numbers 57 214. Let us
Page 75 and 76: The complex numbers 59 219. Prove t
Page 77 and 78: The complex numbers 61 functions (s
Page 79 and 80: 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
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The complex numbers 87 property is
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The complex numbers 89 means will b
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The complex numbers 91 For example,
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The complex numbers 93 of the Riema
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The complex numbers 95 in Figure 38
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The complex numbers 97 permutation
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The complex numbers 99 2.13 Monodro
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The complex numbers 101 First we pr
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The complex numbers 103 and prove t
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Chapter 3 Hints, Solutions, and Ans
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Solutions 107 positive integer unde
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Solutions 109 on the left and by on
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Solutions 111 39. a) See Table 8; b
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Solutions 113 of the real numbers i
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Solutions 115 to all and therefore
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Solutions 117 is a subgroup of the
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Solutions 119 hypothesis. The group
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Solutions 121 The given group is th
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Solutions 123 of symmetries of the
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Solutions 125 i.e., there exists an
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Solutions 127 Answer. The normal su
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Solutions 129 In this way the quoti
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Solutions 131 sends every into itse
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Solutions 133 ement of the commutan
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Solutions 135 143. Let be the three
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Solutions 137 150. See 57. Suppose
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Solutions 139 subgroup. Hence if a
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Solutions 141 is commutative. Since
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Solutions 143 180. Since the lower
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Solutions 145 FIGURE 43 190. The pe
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Solutions 147 permutations of type
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Solutions 149 is a field. 195. We h
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Solutions 151 necessarily that and
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Solutions 153 Let C be the field of
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Solutions 155 in as numerators and
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Solutions 157 c) d) e) where 226. W
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Solutions 159 function of real argu
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Solutions 161 i.e., Consider the re
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Solutions 163 Choose as the smalles
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Solutions 165 Since the function is
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Solutions 167 FIGURE 55 FIGURE 56 F
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Solutions 169 FIGURE 60 257. Suppos
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Solutions 171 265. 266. 267. If the
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Solutions 173 following way. If the
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Solutions 175 for all such that and
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Solutions 177 b) In order for to va
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Solutions 179 to the condition coin
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Solutions 181 are continuous functi
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Solutions 183 turn around the other
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Solutions 185 sought is shown in Fi
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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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Page 277 and 278:
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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