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Groups : a path to geometry

Groups : a path to geometry (1회 대출)

자료유형
단행본
개인저자
Burn, R. P.
서명 / 저자사항
Groups : a path to geometry / R.P. Burn.
발행사항
Cambridge [Cambridgeshire] ;   New York :   Cambridge University Press,   c1985.  
형태사항
xii, 242 p. : Ill., Genealogical tables ; 24 cm.
ISBN
0521300371
일반주기
Includes index.  
서지주기
Bibliography: p. [236]-237.
일반주제명
Group theory. Transformation groups. Geometry.
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001 000000902826
005 19981217143421.0
008 840913s1985 enkaj b 00110 eng
010 ▼a 84021354 //r90
020 ▼a 0521300371
040 ▼a DLC ▼c DLC ▼d DLC ▼d 244002
049 0 ▼l 452045498
050 0 0 ▼a QA171 ▼b .B93 1985
082 0 0 ▼a 512/.22 ▼2 19
090 ▼a 512.2 ▼b B963g
100 1 ▼a Burn, R. P.
245 1 0 ▼a Groups : ▼b a path to geometry / ▼c R.P. Burn.
260 ▼a Cambridge [Cambridgeshire] ; ▼a New York : ▼b Cambridge University Press, ▼c c1985.
300 ▼a xii, 242 p. : ▼b Ill., Genealogical tables ; ▼c 24 cm.
500 ▼a Includes index.
504 ▼a Bibliography: p. [236]-237.
650 0 ▼a Group theory.
650 0 ▼a Transformation groups.
650 0 ▼a Geometry.

소장정보

No. 소장처 청구기호 등록번호 도서상태 반납예정일 예약 서비스
No. 1 소장처 세종학술정보원/과학기술실/ 청구기호 512.2 B963g 등록번호 452045498 도서상태 대출가능 반납예정일 예약 서비스 B M

컨텐츠정보

목차

CONTENTS
Preface = xi
Acknowledgements = xii
1 Functions = 1
  Those properties of functions (or mappings) are established by virtue of which certain sets of functions form groups under composition
  Summary = 7
  Historical note = 8
  Answers = 9
2 Permutations of a finite set = 11
  The cycle notation is developed and even and odd permutations are distinguished
  Summary = 19
  Historical note = 19
  Answers = 20
3 Groups of permutations of R and C = 23
  Every isometry of the plane is shown to be representable as a transformation of C of the form z → e1Θ z + c or of the form z→ e1Θ z ? + c
  Summary = 33
  Historical note = 33
  Answers = 35
4 The M o ·· bius group = 40
  The group of cross-ratio-preserving transformations is shown to be the set of t ransformations of the form z → az+b cz+d , where ad - bc ≠ 0
  Each such transformation is shown to preserve the set of lines and circles in the plane
  Summary = 50
  Historical note = 52
  Answers = 53
5 The regular solids = 57
  The groups of rotations of the regular solids are identified with the permutation groups A4 , S4 and A5 
  Summary = 60
  Historical note = 60
  Answers = 61
6 Abstract groups = 62
  Groups arc defined by four axioms. Some groups are seen to he generated by particular sets of elements. Isomorphism of groups is defined
  Summary = 72
  Historical note = 72
  Answers = 74
7 Inversions of the M o ·· bius plane and stereographic projection = 77
  Stereographic projection is used to show that the group generated by inversions of the M o ·· bius plane is isomorphic to the group of circle-preserving transformations of a sphere
  Summary = 84
  Historical note = 84
  Answers = 85
8 Equivalence relations = 88
  The process of classification is given a formal analysis
  Summary = 91
  Historical note = 91
  Answers = 92
9 Cosets = 93
  A subgroup is used to partition the elements of a group. Lagrange's theorem for finite groups, that the order of a subgroup divides the order of the group, is obtained. A one-one correspondence is established between the cosets of a stabiliser and the orbit of the point stabilised
  Summary = 97
  Historical note = 98
  Answers = 99
10 Direct product = 101
  A simple method of using two given groups to construct one new group is defined
  Summary = 103
  Historical note = 103
  Answers = 104
11 Fields and vector spaces = 105
  When the elements of an additive group (with identity 0) are also the elements of a multiplicative group (without 0) and the operations are linked by distributive laws. the set is called a field when the multiplicative group is commutative. When a multiple direct product is formed with the same additive group of a field as each component, and this direct product is supplied with a scalar multiplication from the field, the direct product is called a vector space
  Summary = 110
  Historical note = 110
  Answers = 112
12 Linear transformations = 114
  When two vector spaces have the same field. a structure-preserving function of one to the other can be described with a matrix
  Summary = 116
  Historical note = 117
  Answers = 118
13 The general linear group GL(2, F) = 119
  The structure-preserving permutations of a 2-dimensional vector space are analysed
  Summary = 123
  Historical note = 124
  Answers = 126
14 The vector space V3 (F) = 128
  Scalar and vector products are defined in three dimensions. The meaning and properties of determinants of 3 × 3 matrices are explored
  Summary = 132
  Historical note = 133
  Answers = 134
15 Eigenvectors and eigenvalues = 136
  Vectors mapped onto scalar multiples of themselves under a linear transformation are found and used to construct a diagonal matrix to describe the same transformation. where possible
  Summary = 143
  Historical note = 143
  Answers = 145
16 Homomorphisms = 148
  Those functions of a group to a group which preserve the multiplicative structure are analysed. The subset of elements mapped to the identity is a normal subgroup. Each coset of that normal subgroup has a singleton image
  Summary = 152
  Historical note = 152
  Answers = 153
17 Conjugacy = 155
  When x and g belong to the same group, the elements x and 1 XR are said to be conjugate. Conjugate permutations have the same cycle structure, Conjugate geometric transformations have the same geometric structure. Normal subgroups are unions of conjugacy classes
  Summary = 162
  Historical note = 162
  Answers = 163
18 Linear fractional groups = 167
  The set of transformations of the form x → ax+b cx+d , where ad - bc ≠ 0, is a homomorphic image of the group GL(2, F)
  Summary = 173
  Historical note = 174
  Answers = 175
19 Quaternions and rotations = 178
  Matrices of the form (- z { W z ) with complex entries, are called quaternions. The set of quaternions satisfies all the conditions for a field except that multiplication is not commutative. The mapping of quaternions given by X → R 1 XR acts like a rotation on 3-dimensional real space
  Summary = 182
  Historical note = 183
  Answers = 184
20 Affine groups = 185
  Line- and ratio-preserving transformations are shown to be combinations of translations and linear transformations
  Summary = 188
  Historical note = 188
  Answers = 190
21 Orthogonal groups = l91
  Isometries are shown to be combinations of translations and linear transformations with matrices A such that A · AT = I. Finite groups of rotations in three dimensions are shown to he cyclic, dihedral or the groups of the regular solids
  Summary = 199
  Historical note = 200
  Answers = 201
22 Discrete groups fixing a line = 205
  If G is a group of isometries and T is its group of translations, the quotient group G/T is isomorphic to a group of isometries fixing a point, called the point group of G. If G fixes a line, its point group is either C1 , C2 , D1 or D2 . This provides a basis for identifying the seven groups of this type
  Summary = 209
  Historical note = 210
  Answers = 211
23 Wallpaper groups = 213
  Groups of isometries not fixing a point or a line are shown to contain translations. If there are no arbitrarily short translations, the translation group has two generators. If such a group contains rotations, their order may only be 2, 3, 4 or 6. The possible point groups are then C1 , C2 , C3 , C4 , C5 , C6 , D1 , D2 , D3 , D4 or D6 . This provides a basis for classifying the seventeen possible groups of this type
  Summary = 225
  Historical note = 225
  Answers = 227
Bibliography = 236
Index = 239

관련분야 신착자료

Aluffi, Paolo (2021)