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Burnside W. — Theory of Groups of Finite Order
Burnside W. — Theory of Groups of Finite Order

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Название: Theory of Groups of Finite Order

Автор: Burnside W.


A classic introduction to group theory for nearly a century, this is the book most often cited in texts on group theory for detailed expositions of basic concepts.
After introducing permutation notation and defining group, the author discusses the simpler properties of group that are independent of their modes of representation; composition-series of groups; isomorphism of a group with itself; Abelian groups; groups whose orders are the powers of primes; and Sylow's theorem. Permutation groups and groups of linear substitutions receive an extensive treatment; two chapters are devoted to the graphical representation of groups, and the closing chapter to congruence groups. Forty-five pages of notes at the back of the book offer ample treatment of special topics.

Язык: en

Рубрика: Математика/Алгебра/Теория групп/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1897

Количество страниц: 388

Добавлена в каталог: 26.03.2005

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Предметный указатель
Abel, quoted      46
Abelian group of order $p^m$ and type (1, 1, ..., 1)      58
Abelian group of order $p^m$ and type (1, 1, ..., 1), group of isomorphisms of      243 244
Abelian group of order $p^m$ and type (1, 1, ..., 1), holomorph of      245
Abelian group of order $p^m$ and type (1, 1, ..., 1), number of distinct ways of choosing a set of independent generating operations of      59
Abelian group of order $p^m$ and type (1, 1, ..., 1), number of sub-groups of, whose order is given      60
Abelian group, definition of      46
Abelian group, definition of, existence of independent generating operations of      52
Abelian group, definition of, invariance of the orders of a set of independent generating operations of      54
Abelian group, definition of, sub-groups of      47 48 55—59
Abelian group, definition of, symbol for, of given type      55
Alternating group, definition of      139
Alternating group, definition of, group of isomorphisms of      246
Alternating group, definition of, is simple, except for degree 4      154
Bochert, quoted      153
Bolza, quoted      162
Burnside, quoted      65 97 157 311 317 335 345 352 365 370
Caucht, quoted      90
Cayley, quoted      306
Characteristic series, definition of      232
Characteristic series, definition of, invariance of      233
Characteristic series, definition of, of a group whose order is a power of a prime      233—235
Characteristic sub-group, definition of      232
Characteristic sub-group, groups with no, are either simple or the direct product of simply isomorphic simple groups      232
Chief composition series, or Chief series, definition of      123
Chief composition series, or Chief series, invariance of      123
Cole and Glover, quoted      137
Cole, quoted      196 370
Colour-groups      306—310
Colour-groups, examples of      310
Complete group, definition of      236
Complete group, group of isomorphisms of a simple group of composite order is a      238
Complete group, groups which contain a, self-conjugately are direct products      236
Complete group, holomorph of a cyclical group of odd order is a      241
Complete group, holomorph of an Abelian group of order $p^m$ and type (1, 1, ..., 1) is a      239
Complete group, symmetric group is a, except for degree 6      246
Composite groups, definition of      29
Composite groups, definition of, non-soluble      376—378
Composite groups, definition of, of even order      360—365
Composition factors, definition of      118
Composition-series, definition of      118
Composition-series, examples of      128 129
Composition-series, invariance of      122
Conjugate operations, definition of      27
Conjugate operations, definition of, complete set of      31
Conjugate sub-groups, complete set of      32
Conjugate sub-groups, definition of      29
Conjugate sub-groups, operations common to or permutable with a complete set of, form a self-conjugate sub-group      33
Dedekind, quoted      89
Defining relations for groups of genus unity      301 302
Defining relations for groups of genus zero      291
Defining relations for groups of order $p^2q$      133—137
Defining relations for groups of order pq      100
Defining relations for groups of orders $p^2$, $p^3$, $p^4$      87 88 89
Defining relations for groups whose orders contain no square factor      354
Defining relations for the holomorph of a cyclical group      240 241
Defining relations for the simple group of order 168      305
Defining relations for the symmetric group of degree 5      305
Defining relations of a group, definition of      21
Defining relations, limitation on the number of, when the genus is given      283
Degbee of a substitution group, definition of      138.
Degbee of a substitution group, is a factor of the order, if the group is transitive      140
Direct product of two groups, definition of      40
Direct product of two groups, represented as a transitive group      190
Direct product of two simply isomorphic groups of order n represented as a transitive group of degree n      146 147.
Doubly transitive groups, generally contain simple self-conjugate sub-groups      192
Doubly transitive groups, of degree n and order n(n-1)      155—157
Doubly transitive groups, the sub-groups of, which keep two symbols fixed      212
Doubly transitive groups, with a complete set of triplets      214
Dtcck, quoted      22 44 172 195 255 292 301 305
Dyck’s theorem that a group of order n can be represented as a regular substitution group of degree n      22—24
Factor groups, definition of      38
Factor groups, invariance of      118.
Factor groups, set of, for a given group      118
Forsyth, quoted      280 283
Fractional linear group, analysis of      319—333
Fractional linear group, definition of      311
Fractional linear group, generalization of      334 335
Frobenius and Stickelberger, quoted      46
Frobenius, quoted      39 45 65 71 90 97 108 110 114 166 232 250 251 345 354 358 360
Frobenius’s theorem that, if n is a factor of the order of a group, the number of operations of the group whose orders divide n is a multiple of n      110—112
Galois, quoted      192 334
General discontinuous group with a finite number of generating operations      256
General discontinuous group, relation of special groups to      257 258
Genus of a group, definition of      280
Gierster, quoted      311
Graphical representation, of a cyclical group      260 261 288
Graphical representation, of a general group      262—266
Graphical representation, of a group of finite order      273—278
Graphical representation, of a group of finite order, examples of      269—272 279 288—290 294 296 299 300 303 310
Graphical representation, of a special group      266—269
Group of isomorphisms, contains the group of cogredient isomorphisms self-con jugately      226
Group of isomorphisms, of a cyclical group      239—242
Group of isomorphisms, of an Abelian group of order $p^n$ and type (1, 1, ..., 1)      243 244 311—317 336—338
Group of isomorphisms, of doubly transitive groups of degree $p^n + 1$ and order $\frac12p^n(p^{2n}- 1)$      246—248
Group of isomorphisms, of the alternating group      246
Group, abelian      46
Group, alternating      139
Group, complete      236
Group, continuous, discontinuous, or mixed      13 14
Group, cyclical      25
Group, defining relations of      21
Group, definition of      11 12
Group, dihedral      287
Group, fundamental or generating operations of      21
Group, general      256
Group, genus of a      280
Group, group of isomorphisms of a      223
Group, holomorph of a      228
Group, icosahedral      289
Group, multiplication table of      20 49
Group, octohedral      289
Group, order of      14
Group, quadratic      326
Group, simple and composite      29
Group, soluble      130 .
Group, special      257
Group, symbol for a      27 (see also “Substitution group”)
Group, symmetric      139
Group, tetrahedral      289 .
Groups of genus one      295—302
Groups of genus two      302
Groups of genus zero      286—292
Groups whose order is $p^m$, where p is a prime      61 et seq.
Groups whose order is always contain self-conjugate operations      62
Groups whose order is case in which there is only one sub-group of a given order      72—75
Groups whose order is determination of distinct types of orders $p^2$, $p^3$, $p^4$, where p is an odd prime      81—88
Groups whose order is every sub-group of, is contained self-conjugately in a greater sub-group      65
Groups whose order is number of sub-groups of given order is congruent to unity, mod. p      71
Groups whose order is number of types which contain self-conjugate cyclical sub-groups of order $p^{m-2}$      76—81
Groups whose order is table of distinct types of orders 8 and 16      88 89
Groups whose orders contain, no cubed factor      359
Groups whose orders contain, no squared factor      353 354
Groups whose sub-groups of order $p^a$ are all cyclical      352 353
Heffter, quoted      215
Holder, quoted      38 87 119 137 224 236 246 249 344 353 370 376
Holomorph of a cyclical group      240—242
Holomorph of an Abelian group of order $p^n$ and type (1, 1, ... 1)      245
Holomorph, definition of      228
Homogeneous linear group, composition series of      314—317
Homogeneous linear group, definition of      244
Homogeneous linear group, generalization of      340—342
Homogeneous linear group, represented as a transitive substitution group      336 337
Homogeneous linear group, simple groups denned by      338 339
Hukwitz, quoted      24 280 282
Identical opebation, definition of      12
Imprimitive groups of degree 6      181 182
Imprimitive groups, definition of      171
Imprimitive self-conjugate sub-group of a doubly transitive group      193 194
Imprimitive systems of a regular group      173
Imprimitive systems of any transitive group      176
Imprimitive systems, definition of      172
Imprimitive systems, properties of      185 186
Intbansitive groups, definition of      140
Intbansitive groups, of degree 7 with transitive sets of 3 and 4      163 164
Intbansitive groups, properties of      159—162
Intbansitive groups, transitive sets of symbols in      159 166
Isomoephism of general and special groups      257 258
Isomoephism of two groups, general, definition of      41
Isomoephism of two groups, multiple, definition of      36
Isomoephism of two groups, simple, definition of      32
Isomoephisms of a group with itself, definition of      222
Isomoephisms of a group, class of, definition of      224
Isomoephisms of a group, cogredient and contragredient, definition of      224
Isomoephisms of a group, limitation on the order of      252
Joedan, quoted      46 119 143 146 149 150 153 189 196 198 204 311
Klein and Feicke, quoted      162
Klein, quoted      224 289 292
Limitation on the number of defining relations of a group of given genus      283
Limitation on the order of a group of given genus      282
Maeggeaff, quoted      198
Maillet, quoted      108
Maschke, quoted      310
Mathieu, quoted      155
Maximum self-conjugate sub-group, definition of      35
Maximum sub-group, definition of      35
Millee, quoted      105 196 259
Minimum self-conjugate sub-group is a simple group or the direct product of simply isomorphic simple groups      127
Minimum self-conjugate sub-group, definition of      124
Mooee, quoted      215 335
Multiplication table of a group      20 49
Multiply isomoephic groups, definition of      36
Netto, quoted      141 198 215
Numbee of symbols unchanged by all the substitutions of a group      165 166
Operations common to two groups form a group      27
Operations of a group, which are permutable with a given operation or sub-group, form a group      29 31
Operations, common to or permutable with each of a complete set of conjugate subgroups form a self-conjugate sub-group      33
Order of a group, definition of      14
Order of an operation, definition of      14
Permutability of an operation with a group, definition of      29
Permutable groups, definition of      41
Permutable opeeations, definition of      12
Primitive groups, definition of      171
Primitive groups, limit to the order of, for a given degree      199
Primitive groups, of degree 6      205 206
Primitive groups, of degree 7      206—208
Primitive groups, of degree 8      209—211
Primitive groups, of degrees 3, 4 and 5      205
Primitive groups, of prime degree      201
Primitive groups, when soluble, have a power of a prime for degree      192
Primitive groups, with a transitive sub-group of smaller degree      197 198
Primitivity, test of      184
Regular division of a surface, representation of a group by means of      278
Repeesentation, graphical      see “Graphical representation”
Representation of a group, in primitive form      177
Representation of a group, in transitive form. i.e. as a transitive substitution group      22 173—179
Self-conjugate operation, definition of      28
Self-conjugate operation, of a group whose order is the power of a prime      62
Self-conjugate operation, of a transitive substitution group must be a regular substitution      144
Self-conjugate sub-group, definition of      29
Self-conjugate sub-group, generated by a complete set of conjugate operations      34
Self-conjugate sub-group, of a k-ply transitive group is, in general, (k-1)-ply transitive      189
Self-conjugate sub-group, of a primitive group must be transitive      187
Self-conjugate sub-group, of an imprimitive group      187 188
Simple groups, definition of      29
Simple groups, whose orders are the products of not more than 5 primes      367—370
Simple groups, whose orders do not exceed 660      370—375
Simply isomorphic groups, definition of      22
Simply isomorphic groups, said to be of the same type      22
Soluble groups, definition of      130
Soluble groups, properties of      131 132
Soluble groups, special classes of      345—359
Sub-group, definition of      25
Sub-group, order of a, divides order of group      26 (see also “Conjugate” “Self-conjugate”
Substitution group, conjugate substitutions of, are similar      144
Substitution group, construction of multiply transitive      150
Substitution group, degree of transitive, is a factor of its order      140
Substitution group, degree of, definition of      138
Substitution group, doubly transitive, of degree n and order n(n-l)      155—157
Substitution group, limit to the degree of transitivity of      152
Substitution group, multiply transitive, definition of      148
Substitution group, order of a ft-ply transitive, whose degree is n, is a multiple of n(n-1)...(n-k+1)      148
Substitution group, primitive and imprimitive, definition of      171
Substitution group, quintuply transitive, of degree 12      170
Substitution group, regular, definition of      24
Substitution group, representation of any group as a regular      22
Substitution group, transitive and intransitive, definition of      140
Substitution group, transitive, whose substitutions, except identity, displace all or all but one of the symbols      141—144
Substitution group, triply transitive, of degree n and order n(n-l)(n-2)      158
Substitution groups whose orders are powers of primes      218 219
Substitution, circular      7
Substitution, cycles of      2
Substitution, definition of      1
Substitution, even and odd      10
Substitution, identical      4
Substitution, inverse      4
Substitution, order of      6
Substitution, permutable      8
Substitution, regular      7
Substitution, similar      7
Substitution, symbol for the product of two or more      4
Substitutions which are permutable, with a given substitution      215 216
Substitutions with every substitution of a given group      145 146 217
Substitutions with every substitution of a group, whose degree is equal to its order, form a simply isomorphic group      146
Sylow, quoted      62 90
Sylow’s Theorem      92—95
Sylow’s theorem, extension of      110
Sylow’s theorem, some direct consequences of      97—100
Symbol for a group generated by given operations      27
Symbol for the product of two or more operations      12
Symbol for the product of two or more substitutions      4
Symbol, $\vartheta(P)$ and $\theta(P)$, definition of      250 251
Symmetric group, definition of      139
Symmetric group, is a complete group, except for degree 6      246
Symmetric group, of degree 6 has 12 simply isomorphic sub-groups of order 5! which form two distinct conjugate sets      200
Symmetric group, of degree n has a single set of conjugate sub-groups of order (n-1)! except when n is 6      200
Transforming an operation, definition of      27
Transitive group, definition of      140
Transitive group, number of distinct modes of representing the alternating group of degree 5 as a      179 180
Transitive group, representation of any group as a      175—179
Transpositions, definition of      9
Transpositions, number of, which enter in the representation of a substitution is either always even or always odd      10
Transpositions, representation of a substitution by means of      9
Triply transitive groups of degree n and order n(n-l)(n-2)      158
type of a group      see “Simply isomorphic groups”
Types of group      8 or 16 88 89
Types of group 24      101—104
Types of group 60      105—108
Types of group, $p^2q$, where p and q are different primes      133—137
Types of group, distinct, whose order is $p^2$, $p^3$, or $p^4$, where p is an odd prime      87 88
Types of group, pq, where p and q are different primes      100 101
Weber, quoted      46
Young, quoted      63 68 87 310
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