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W. Burnside — Theory of Groups of Finite Order

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Название: Theory of Groups of Finite Order

Автор: W. Burnside

Аннотация:

Very considerable advances in the theory of groups of finite order have been made since the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 1911

Количество страниц: 542

Добавлена в каталог: 15.04.2013

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Предметный указатель

Abel      99
Abelian group      24
Abelian group of order $p^{n}$ and type (1, 1,..., 1), group of isomorphisms of      89
Abelian group of order $p^{n}$ and type (1, 1,..., 1), holomorph of      90
Abelian group of order $p^{n}$ and type (1, 1,..., 1), number of sub-group of given order of      84
Abelian group of order $p^{n}$ and type (1, 1,..., 1), number of ways of choosing a set of independent generating operations of      85
Abelian group, characteristic series of      82
Abelian group, existence of independent generating operations of      77
Abelian group, is direct product of groups whose orders are powers of primes      75
Abelian group, orders of the isomorphisms of      86
Abelian group, sub-groups of      78—81 84
Abelian group, symbol for, of given type      80
Alternating group      132
Alternating group of degree 5, represented as an irreducible group in 3 symbols      232
Alternating group, group of isomorphisms of      162
Alternating group, is simple except for degree      4 139
Baker      358
Blickfeldt      348 352
Bochert      180
Bolza      188
Burkhardt      485
Burnside      91 122 163 166 184 243 269 306 311 323 328 353 452 464 468 485
Castelnuovo      360
Cauchy      47
Cayley      423
Central of a group      27
Characteristic equation of a linear substitution      192
Characteristic of a linear substitution      192
Characteristic series      69
Characteristic series of an Abelian group      82
Characteristic series, invariance of factor-groups of      69
Characteristic sub-group      68
Characteristic sub-group, groups with no, are either simple or the direct product of simply isomorphic simple groups      68
Chief composition-series      51
Chief composition-series or chief-series, examples of      55 56
Chief composition-series or chief-series, invariance of factor-groups of      51
Circular permutation      9
Cole      504
Cole and Glover      80
Colour group      305
Commutator      38
Commutator sub-group      39
Complete group      70
Complete group, group of isomorphisms of a simple group of composite order is a      71
Complete group, groups which contain a, self-conjugately are direct products      70
Complete group, if an Abelian group of odd order is a characteristic sub-group of its holomorph, the latter is a      72
Complete group, symmetric group is a, except for degree      6 162
Completely reduced form of a transitive group      207
Completely reducible group      197
Composite group      24
Composition of two groups of linear substitutions      191
Composition-factors      48
Composition-series      48
Composition-series, examples of      55 56
Composition-series, invariance of factor-groups of      50
Conjugate groups of linear substitutions      190
Conjugate groups of linear substitutions, are simply isomorphic      190
Conjugate groups of linear substitutions, possess an invariant Hermitian form      195
Conjugate operations      24
Conjugate set of operations      26
Conjugate sets, invariant property of multiplication table of      236
Conjugate sets, multiplication of      41—46
Conjugate sub-groups      24
Cycle of a permutation      4
Cyclical group      22
Cyclical group, group of isomorphisms of      88
Cyclical group, holomorph of      88
Dedekind      146
Defining relations of a group for groups of genus one      302
Defining relations of a group for groups of genus two      302
Defining relations of a group for groups of genus zero      296
Defining relations of a group for groups of order      24 126
Defining relations of a group for groups of order $p^{2}$ , $p^{3}$ , $p^{4}$       117
Defining relations of a group for groups of order $p^{2}q$       59
Defining relations of a group for groups of order pq      36
Defining relations of a group for groups whose Sylow sub-groups are cyclical      129
Defining relations of a group for the holomorph of a cyclical group      88
Defining relations of a group for the simple group of order      168 303
Defining relations of a group, limitations on the number of, when the genus is given      291
Degree of a permutation-group      131
Derived group      39
Derived groups, properties of      39
Derived groups, series of      40
Determinant of a linear substitution      188
Dickson      428 485 504
Dihedral group      295
Direct product of two groups      31
Direct product, of two groups represented as a transitive group      152
Direct product, of two simply isomorphic groups of order n without self-conjugate operations represented as a transitive group of degree n      64 136
Distinct representations      174 204
Doubly transitive group has two irreducible components      250
Doubly transitive group of degree $p^{n}$ and order $p^{n}(p^{n}-1)$       140
Doubly transitive group with a complete set of triplets      169
Doubly transitive group, is in general simple or contains a simple self-conjugate sub-group      154
Doubly transitive group, the sub-groups of, which keep two symbols fixed      167
Dyck      44 231 372 383 401 409 418 422
Elliott      289
Equivalent representations      174 204
Even permutation      11
Factor-group      29
Family of irreducible representations      234
Forsyth      397 400
Fractional linear group      314
Fractional linear group, analysis of      315—327
Fractional linear group, generalization of      328
Frobenius      39 45 49 53 92 93 108 122 129 149 153 173 269 328 331 334 502
Galois      202 451
Generalized linear homogeneous group      882
Genus of a group      289
Gierster      428
Graphical representation, of a cyclical group      276
Graphical representation, of a general group      277—280
Graphical representation, of a group of finite order      287 288 291
Graphical representation, of a special group      281—286
Graphical representation, of groups of genus one      298—301
Graphical representation, of groups of genus zero      295
Graphical representation, of the simple group of order      168 303
Group      12
Group of isomorphisms      62
Group of isomorphisms of a cyclical group      88
Group of isomorphisms of au Abelian group of order $p^{n}$ and type (1, 1, ..., 1)      89
Group of isomorphisms of the alternating group      162
Group of isomorphisms, general properties of      62—67
Group of monomial substitutions      242
Group-characteristics, calculation of      222 223
Group-characteristics, congruences between      256 257
Group-characteristics, relations between      211—216 218 219
Groups of linear substitutions of finite order are irreducible or completely reducible      200
Groups of linear substitutions, composition of      191
Groups of linear substitutions, general properties of      188
Groups of linear substitutions, self-conjugate operations of      202 203
Groups of linear substitutions, standard form of      196
Groups of order $p^{\alpha}q^{\beta}$ , characteristic sub-groups of      241
Groups of order $p^{\alpha}q^{\beta}$ , where p and q are primes, are simple      240
Groups whose order is $p^{n}$ are distinct from their derived groups      94
Groups whose order is $p^{n}$ which contain a cyclical sub-group of order $p^{n-2}$ self-conjugately      109—111
Groups whose order is $p^{n}$ with only one sub-group of given order      104 105
Groups whose order is $p^{n}$ , always have self-conjugate operations      92
Groups whose order is $p^{n}$ , every sub-group of, is contained self-conjugately in a sub-group of greater order      96
Groups whose order is $p^{n}$ , irreducible representations of, can be expressed as monomial groups      258
Groups whose order is $p^{n}$ , number of sub-groups of given order is congruent to 1 (mod. p)      103
Groups whose order is $p^{n}$ , types of, when n = 2, 3, 4      112—117
Groups whose order is $p^{n}$ , where p is prime, general properties of      92
Groups whose Sylow sub-groups are cyclical      128 129
Heffter      224
Hermitian form      194
Hermitian forms, invariant for group and its conjugate      195
Hermitian forms, number of, for given group of linear substitutions      206
hilbert      606
Hilton      101 427

Hoelder      39 65 80 84 94 95 146 166 209 210 604
Holomorph of a cyclical group      88
Holomorph of a group      64
Holomorph of an Abelian group of order $p^{n}$ and type (1, 1, ..., 1)      90
Holomorph of any Abelian group      87
Holomorph, general properties of      64
Homogeneous linear group, composition-series of      811—813
Homogeneous linear group, general properties of      308—310
Homogeneous linear group, generalization of      332 333
Homogeneous linear group, represented as a doubly transitive group      329
Hurwitz      397 399
Icosahedral group      295
Identical isomorphism      61
Identical operation      13
Identical representation      205
Imprimitive group      146
Imprimitive self-conjugate sub-group of a doubly transitive group      151
Imprimitive systems of a regular permutation-group      176
Imprimitive systems of any transitive group      177
Imprimitive systems, properties of      147 148
Inner isomorphism      63
Intransitive group      133
Intransitive groups, general properties of      142—144
Intransitive groups, test for      145
Intransitive groups, transitive constituents of      142
Invariant of a group of linear substitutions      260
Invariants of a group of linear substitutions of an irreducible group      269
Invariants of a group of linear substitutions, absolute and relative invariants      260
Invariants of a group of linear substitutions, examples of      266—268
Invariants of a group of linear substitutions, quadratic      270
Invariants of a group of linear substitutions, system of, in terms of which all are rationally expressible      263
Inverse conjugate sets      41
Inverse of an operation      12
Irreducible components of a group of linear substitutions      205
Irreducible group of linear substitutions      197
Isomorphism      61
Isomorphisms, limitation on the order of      86
Isomorphisms, of a general and a special group      275
Isomorphisms, of a group with itself, inner and outer      63
Isomorphisms, which change every conjugate set into itself      65 249
Isomorphisms, which leave no operation except E unchanged      66 248
JORDAN      22 65 99 172 176 178 180 198 204 207 428 455 485
Klein      407
Klein and Fricke      188
Linear homogeneous group      90
Linear substitution      188
Loewy      243 256 266
Maclagan-Wedderburn      45
Maillet      163
Marggraff      207
Mark of a sub-group of a permutation-group      180
Maschke      243 427
Mathieu      182
Maximum self-con jugate sub-group      27
Maximum sub-group      27
Metabelian group      40
Miller      98 134 147 161 503
Minimum self-conjugate sub-group      52
Minimum self-conjugate sub-group, is a simple group or the direct product of simply isomorphic simple groups      53
Minkowski      484 506
Molien      300
Moore      224 256 452 464 468
Multiplication table of conjugate sets      44
Multipliers of a linear substitution      193
Multiply isomorphic groups      28
Multiply transitive group      137
Multiply transitive groups, examples of      141
Multiply transitive groups, general properties of      137 138
Multiply transitive groups, self-conjugate sub-groups of      150
Netto      171 207 224
Number of operations whose nth powers are conjugate to a given operation      37
Octohedral group      295
Odd permutation      11
Order of a group      15
Order of a permutation      8
Outer isomorphism      63
Permutable groups      33
Permutable groups, general properties of      34 35
Permutable operations      12
Permutation      2
Permutation-group, doubly transitive, of degree $p^{n}$ and order $p^{n}(p^{n}-1)$       140
Permutation-group, general properties of      131
Permutation-group, limits to the degree of transitivity      138 160
Permutation-group, order of k-ply transitive, whose degree is n, is a multiple of n(n - 1)...(n - k + 1)      137
Permutation-group, quintuply transitive, of degree      12 173
Permutation-group, representation of a group as      174
Permutation-group, transitive, whose order is the power of a prime      173
Permutation-group, transitive, whose permutations displace all or all but one of the symbols      134 247
Permutation-group, triply transitive, of degree $p^{n} + 1$ and order $p^{n}(p^{2n}-1)$       141
Permutations which are permutable, with a given permutation      170
Permutations which are permutable, with every permutation of a given group      171
Permutations which are permutable, with every permutation of a regular permutation-group      20 136
Primitive group      146
Primitive groups of degrees 3 to 8      166
Primitive groups with transitive sub-group of smaller degree      158 159
Primitive groups, general properties of      147 177
Primitive groups, limit to order of, for given degree      160
Primitive groups, simply transitive, of prime degree, are soluble      251
Primitive groups, test for      147
Primitive groups, when soluble have the power of a prime for degree      154
Quadratic group      321
Quaternion group      106
Reduced variables      205
Reducible group of linear substitutions      197
Regular permutation      9
Relative invariant of a group of linear substitutions      260
Representation of a group as a group of linear substitutions      204
Representation of a group as a group of monomial substitutions      242
Representation of a group as a permutation-group      174
Representation of a group, as a permutation-group      174
Rietz      603
schur      311
Self-conjugate operation      24
Self-conjugate operation, of a transitive permutation-group      135
Self-conjugate operation, of an irreducible group of linear substitutions      202
Self-conjugate sub-group      24
Self-conjugate sub-group, determined by congruences between the group-characteristics      256 257
Self-conjugate sub-group, generated by a complete set of conjugate operations      27
Self-conjugate sub-group, of a primitive group must be transitive      149
Self-conjugate sub-group, of an imprimitive group      146 148
Self-conjugate sub-group, of an intransitive group      142
Series of derived groups      40
Set of generating operations      18
Set of group-characteristics      212
Similar permutations      10
Simple group      24
Simple group of order 168 represented as an irreducible group in 3 variables      232
Simple groups, group of isomorphisms of, is complete      71
Simple groups, order if even is divisible by 12, 16 or 56      245
Simple groups, systems of      139 328 329 332
Simply isomorphic groups      19
Simply isomorphic groups, concrete examples of      17
Soluble group      40 68
Sommer      506
Sub-group      22
Sylow      119 149
Sylow's theorem      120
Sylow's theorem, deductions from      122—125
Sylow's theorem, generalization of      121
Symmetric group      132
Symmetric group, is a complete group, except for degree      6 162
Tetrahedral group      295
Transitive group      133
Transposed groups of linear substitutions      190
Transposition      11
Transpositions, number of, which enter in a permutation is either always even or always odd      11
Transpositions, representation of a permutation by means of      11
Type ( $m_{1}, m_{2},..., m_{s}$ ) of Abelian group      80
Weber      99 269 506
Western      80

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