Theory of groups of finite order / by W. Burnside.
Editor: [New York?] : Dover, 1955Edición: 2nd edDescripción: xxiv, 512 p. ; 21 cmOtra clasificación: 20CONTENTS CHAPTER I. ON PERMUTATIONS. PAGE Object of the chapter [1] Definition of a permutation [1] Notation for permutations; cycles; products of permutations [1] Identical permutation; inverse permutations; order of a permutation [4] Circular, regular, similar and permutable permutations [7] Transpositions; representation of a permutation as a product of transpositions; odd and even permutations; Examples [9] CHAPTER II. THE DEFINITION OF A GROUP. Definition of a group [11] The identical operation [12] Continuous, mixed, and discontinuous groups [13] Order of an operation; product of operations; every operation of order mn, m and n relatively prime, can be expressed in just one way as the product of permutable operations of orders m and n [14] Examples of groups of operations; multiplication table of a group [17] Generating operations of a group; defining relations; simply isomorphic groups [20] Representation of a group of order N as a group of regular permutations of A symbols [22] Various modes of representing groups [24] CHAPTER III. ON THE SIMPLER PROPERTIES OF A GROUP WHICH ARE INDEPENDENT OF ITS MODE OF REPRESENTATION. Sub-groups; the order of a sub group is a factor of the order of the group containing it; various notations connected with a group and its sub-groups [25] Common sub-group of two groups; further notations. [27] Transforming one operation by another; conjugate operations and sub-groups; self-conjugate operations and self-con jugate sub-groups; Abelian groups; simple and composite groups [29] The operations of a group which are permutable with a given operation or sub-group form a group [31] Complete conjugate sets of operations and sub-groups [33] Theorems concerning self-conjugate sub-groups; maximum self-conjugate sub-groups; maximum subgroups [34] Multiply isomorphic groups; factor-groups; direct product of two groups [37] General isomorphism between two groups [41] Permutable groups; the group generated by two selfconjugate sub-groups of a given group; Examples [42] CHAPTER IV. FURTHER PROPERTIES OF A GROUP WHICH ARE INDEPENDENT OF ITS MODE OF REPRESENTATION. 35 If pm(p prime) divides the order of a group, there is a sub-group of order pm [46] 36 Groups of order p2 and pq [48] 37 The number of operations of a group of order N whose nth powers are conjugate to a given operation is zero or a multiple of the highest common factor of N and n [49] 38—40 Commutators; commutator sub-group or derived group; series of derived group; soluble groups; metabelian groups [54] 41—43 Multiplication of conjugate sets; inverse sets; selfinverse set [57] 44—47 Multiplication table of conjugate sets; deductions from it [60] CHAPTER V. ON THE COMPOSITION-SERIES OF A GROUP 48 The composition-series, composition-factors and factorgroups of a given group [64] 49, 50 Invariance of the factor-groups for different composition-series [65] 51 Chief composition-series; invariance of its factor-groups [68] 52, 53 Nature of the factor-groups of a chief-series; minimum self-conjugate sub-groups [69] 54 Construction of a composition-series to contain a given chief-series [71] 55, 56 Examples of composition-series [72] 57, 58 Theorems concerning composition-series [74] 59 Groups of order p2q [76] CHAPTER VI. ON THE ISOMORPHISM OF A GROUP WITH ITSELF. 60 Object of the chapter [81] 61 Definition of an isomorphism; identical isomorphism. [82] 62 The group of isomorphisms of a group 82, [83] 63 Inner and outer isomorphisms; the inner isomorphisms constitute a self-conjugate sub-group of the group of isomorphisms [84] 64 The holomorph of a group [86] 65, 66 Isomorphisms which permute the conjugate sets [88] 67 Permutation of sub-groups by the group of isomorphisms [91] 68 Definition of a characteristic sub-group; nature of a group with no characteristic sub-group. [92] 69 Characteristic-series of a group [93] 70 Definition of a complete group; a group with a complete group as a self-conjugate sub-group must be a direct product [93] 71, 72 Theorems concerning complete groups [95] 73 The orders of certain isomorphisms; Examples [97] CHAPTER VII. ON ABELIAN GROUPS. 74 Introductory [99] 75 Every Abelian group is the direct product of Abelian groups whose orders are powers of primes [100] 76 Limitation of the discussion to Abelian groups whose orders are powers of primes [101] 77 Existence of a set of independent generating operations for such a group [101] 78 The orders of certain sub-groups of such a group [103] 79, 80 Invariance of the orders of the generating operations ; simply isomorphic Abelian groups; symbol for Abelian group of given type [104] 81 Determination of all types of sub-group of a given Abelian group [106] 82 Characteristic series of an Abelian group [108] 83, 84 Properties of an Abelian group of type (1, 1, ..., 1) [110] 85 The group of isomorphisms and the holomorph of such a group [111] 86 The orders of the isomorphisms of an Abelian group [112] 87 The group of isomorphisms and the holomorph of any Abelian group [113] 88 The group of isomorphisms and the holomorph of a cyclical group 114, [115] 89, 90 The linear homogeneous group; Examples 116—119 CHAPTER VIII. ON GROUPS WHOSE ORDERS ARE THE POWERS OF PRIMES. 91 Object of the chapter [119] 92 Every group whose order is the power of a prime contains self-conjugate operations . [119] 93 The series of self-conjugate sub-groups H1, H2, ..., Hn ,E, such that is the central of G|H i+1 [120] 94, 95 The series of derived groups [120] 96 Every sub-group is contained self-conjugately in a sub-group of greater order [122] 97 The operations conjugate to a given operation [123] 98, 99 Illustrations of preceding paragraphs [124] 100 Operations conjugate to powers of themselves [126] 101—103 Number of sub-groups of given order is congruent to 1, mod. p [128] 104 Groups of order pm with a single sub-group of order p8 are cyclical, where p is an odd prime [130] 105 Groups of order 2m with a single sub-group of order 28 are cyclical unless s is 1, in which case there is just one other type [131] 106 The quaternion group [132] 107 Some characteristic sub-groups [133] 108, 109 Groups of order pm with a self-conjugate cyclical sub-group of order pm-1 [134] 110, 111 Groups of order pm with a self-conjugate cyclical sub-group of order pm-2 [136] 112 Distinct types of groups of orders p2 and p3 [139] 113—116 Distinct types of groups of order p4 [140] 117, 118 Tables of groups of orders p2, p3 and p4 [144] 119 Examples [146] CHAPTER IX. ON SY LOW’S THEOREM. 120 Proof of Sylow’s theorem [149] 121 Generalisation of Sylow’s theorem [152] 122 . Theorem concerning the maximum common subgroup of two Sylow sub-groups [153] 123—125 Further theorems concerning Sylow sub-groups [154] 126 Determination of all distinct types of group of order 24 [167] 127 Determination of the only group of order 60 with no self-conjugate sub-group of order 5 [161] 128, 129 Groups whose Sylow sub-groups are all cyclical; their defining relations [163] 130 Groups with properties analogous to those of groups whose orders are powers of primes; Examples [166] CHAPTER X. ON PERMUTATION-GROUPS : TRANSITIVE AND INTRANSITIVE GROUPS: PRIMITIVE AND IMPRIMITIVE GROUPS. 131 The degree of a permutation-group [168] 132 The symmetric and the alternating groups . [169] 133 Transitive and intransitive groups; the degree of a transitive group is a factor of the order . 170, [171] 134 Transitive groups whose permutations, except identity, permute all or all but one of the symbols [171] 135 Conjugate permutations are similar; self-conjugate operations and self-conjugate sub-groups of a transitive group [173] 136 Transitive groups of which the order is equal to the degree [174] 137 Multiply transitive groups; the order of a k-ply transitive group of degree n is divisible by re(n—1) ...(n —k+1) [176] 138 Groups of degree n, which do not contain the alternating group, cannot be more than ((1/3)n+l)-ply transitive [178] 139 The alternating group of degree re is simple except when n is 4 [180] 140, 141 Examples of doubly and triply transitive groups . [181] 142—144 Intransitive groups; transitive constituents; the general isomorphisms between two groups [186] 145 Tests of transitivity [189] 146 Definition of primitivity and imprimitivity; imprimitive systems [191] 147 Test of primitivity [192] 148 Properties of imprimitive systems .... [194] 149 Self-conjugate sub-groups of transitive groups; a self-conjugate sub-group of a primitive group must be transitive [195] 150 Self-conjugate sub-groups of ir-ply transitive groups are in general (k —l)-ply transitive [197] 151—154 Further theorems concerning self-conjugate subgroups of multiply transitive groups; a group which is at least doubly transitive must, in general, either be simple or contain a simple group as a self-conjugate sub-group [198] 155 Construction of a primitive group with an imprimitive self-conjugate sub-group . [202] 156 Examples [203] CHAPTER XI. ON PERMUTAT1ON-GROUPS : TRAN8ITIVITY AND PR1MITIVITY: (concluding properties). 157—160 Primitive groups with transitive sub-groups of smaller degree; limit to the order of a primitive group of given degree [205] 161 Property of the symmetric group . . . . [208] 162 The symmetric group of degree n is a complete group except when n is 6; the group of isomorphisms of the symmetric group of degree 6 [209] 163—165 Further limitations on the order of a primitive group; examples of the same [210] 166 Determination of all primitive groups whose degrees do not exceed 8 [214] 167—169 Sub-groups of doubly transitive groups which leave two symbols unchanged; complete sets of triplets [221] 170, 171 The most general permutation-group each of whose operations is permutable with a given permutation, or with every permutation of a given group [224] 172 The most general transitive group whose order is the power of a prime [227] 173 Examples [229]
CHAPTER XII. ON THE REPRESENTATION OF A GROUP OF FINITE ORDER AS A PERMUTATION-GROUP. 174 Definition of representation; equivalent and distinct representations [231] 175 The two representations of a group as a regular permutation-group given by pre- and postmultiplication are equivalent [232] 176 The imprimitive systems in the representation of a group as a regular permutation-group [232] 177—179 To each conjugate set of sub-groups there corresponds a transitive representation; every transitive representation arises in this way [233] 180, 181 The mark of a sub-group in a representation; table of marks; there are just s distinct representations, where s is the number of distinct conjugate sets of sub-groups [236] 182, 183 The same set of permutations may give two or more distinct representations ; connection with outer isomorphisms [239] 184, 185 Composition of representations; number of representations of given degree [240] 186 A more general definition of equivalence [241] 187 Alternative process for setting up representations [242] CHAPTER XIII. ON GROUPS OF LINEAR SUBSTITUTIONS J REDUCIBLE AND IRREDUCIBLE GROUPS. 188, 189 Linear substitutions; their determinants; groups of linear substitutions [243] 190 Transposed groups of linear substitutions; conjugate groups of linear substitutions; generalisation [245] 191 Composition of isomorphic groups of linear substitutions [247] 192 Characteristic equation of a substitution; characteristic of a substitution [249] 193 Canonical form of a linear substitution of finite order [251] 194 Definition of an Hermitian form; definite forms; properties of a definite form [253] 195 Existence of a definite Hermitian form which is invariant for a group and its conjugate. [255] 196 Standard form for a group of linear substitutions of finite order [256] 197 Reducible and irreducible groups of linear substitutions ; completely reducible groups [258] 198—200 A group of linear substitutions of finite order is either irreducible or completely reducible [259] 201 Proof of the preceding result when the coefficients are limited to a given algebraic field [264] 202 Substitutions permutable with every substitution of an irreducible group [265] 203 The group of linear substitutions permutable with every substitution of a given group of linear substitutions; Examples; Note [266] CHAPTER XIV. ON THE REPRESENTATION OF A GROUP OF FINITE ORDER AS A GROUP OF LINEAR SUBSTITUTIONS. Definition of a representation; distinct and equivalent representations; Examples [269] The identical representation; irreducible components of a representation; reduced variables [271] The number of linearly independent invariant Hermitian forms for a representation and its conjugate [272] The completely reduced form of any representation of a group as a transitive permutation-group [273] The completely reduced form of the representation of a group as a regular permutation-group; all the irreducible representations occur in it; the number of distinct irreducible representations is equal to the number of conjugate sets [276] Irreducible representations with which the group is multiply isomorphic; irreducible representations in a single symbol [278] CHAPTER XV. ON GROUP-CHARACTERISTICS. 211 Explanation of the notation [280] 212 Set of group-characteristics; in conjugate representations corresponding group-characteristics are conjugate imaginaries [281] Proof of relations between the sets of group-characteristics [283] Two representations of a group are equivalent if, and only if, they have the same group-characteristics [287] Relations between the representation of a group as a transitive permutation-group when the more general definition of equivalence is used [288] Further relations between the group-character-istics; table of relations [290] Composition of the irreducible representations Two distinct conjugate sets cannot have the same characteristic in every representation [293] Case of groups of odd order [294] Determination of the characteristics from the multiplication table of the conjugate sets; Example [295] The number of variables operated on by an irreducible representation is a factor of the order of the group [297] Property of set of irreducible representations which combine among themselves by composition [298] Completely reduced form of the group on the homogeneous products of the variables operated on by a group of linear substitutions [300] The irreducible representation and conjugate sets of a factor-group [301] The reduction of a regular permutation-group ; the complete reduction of the general group {G, G'} of § 136 [302] The representation of the simple groups of orders 60 and 168 as irreducible groups in 3 variables [307] Nature of the coefficients in a group of linear substitutions of finite order [311] Families of irreducible representations; the number of families is equal to the number of distinct conjugate sets of cyclical sub-groups [311] The characteristics of a family of representations Invariant property of the multiplication table of conjugate sets [316] Similar invariant property of the composition table of the irreducible representations [.317] Examples [318] CHAPTER XVI. SOME APPLICATIONS OF THE THEORY OF GROUPS OF LINEAR SUBSTITUTIONS AND OF GROUP-CHARACTERISTICS. §§ PAGE 239 Introductory [321] 240, 241 Groups of order are soluble [321] 242 Representation of a group as a group of monomial substitutions [324] 243 Application of this representation to obtain conditions for the existence of self-conjugate sub-groups [325] 244 Particular cases; a group whose order is not divisible by 12 or by the cube of a prime is soluble [327] 245 Further particular cases; the order, if even, of a simple group is divisible by 12, 16 or 56 [328] 246 Relations between the characteristics of a group and those of any sub-group [330] 247 A transitive permutation-group whose operations permute all or all but one of the symbols has a regular self-conjugate sub-group [331] 248 Groups of isomorphisms which leave E only unchanged [334] 249 Isomorphisms which change each conjugate set into itself [336] 250 The irreducible components of a transitive permutation-group [338] 251 Simply transitive groups of prime degree are soluble [339] 252 Generalisation of preceding theorem [341] 253 On the result of compounding an irreducible group with itself; some properties of groups of odd order [343] 254 Criterion for the existence of operations of composite order [346] 255 On certain Abelian sub-groups of irreducible groups [348] 256, 257 Congruences between characteristics which indicate the existence of self-conjugate sub-groups; illustrations [349] 258 Every irreducible representation of a group whose order is the power of a prime can be expressed as a group of monomial substitutions [351] 259 Examples [353] CHAPTER XVII. ON THE INVARIANTS OF GROUPS OF LINEAR SUBSTITUTIONS. 260, 261 Definition of invariants and relative invariants; condition for existence of relative invariants; invariant in the form of a rational fraction [355] 262 Existence of an algebraically independent set of invariants [357] 263 Formation, for a group in n variables, of a set of n+1 invariants in terms of which all invariants are rationally expressible [357] 264 On the possibility of replacing the above set of n+1 invariants by a set of n [360] 265 The group of linear substitutions for which each of a given set of functions is invariant [360] 266—268 Examples of sets of invariants for certain special groups [362] 269 Property of invariants of an irreducible group [366] 270 Condition that an irreducible group may have a quadratic invariant [367] 271 General remarks on the relation of a group to its invariants [369] 272 Examples [370] CHAPTER XVIII. ON THE GRAPHICAL REPRESENTATION OF A GROUP. Introductory remarks [372] The most general discontinuous group that can be generated by a finite number of operations ; the relation of this group to the special group that arises when one or more relations hold between the generating operations [373] Graphical representation of a cyclical group [376] Graphical representation of the general group, when no relations connect the generating operations [379] Graphical representation of the special group when relations connect the generating operations [384] Illustration of the preceding paragraphs [386] Graphical representation of the special group when the generating operations are of finite order, [389] 287, 288 Graphical representation of a group of finite order [394] 289 The genus of a group [397] 290, 291 Limitation on the order and on the number of defining relations of a group of given genus: Examples; Note [398] CHAPTER XIX. ON THE GRAPHICAL REPRESENTATION OF GROUPS: GROUPS OF GENUS ZERO AND UNITY: CAYLEY’S COLOUR-GROUPS. 292 The diophantine relation connecting the order, the genus, and the number and orders of the generating operations [402] 293—296 Groups of genus zero : their defining relations and graphical representation; the dihedral, tetrahedral, octahedral and icosahedral groups 403—409 297—302 Groups of genus unity: their defining relations and graphical representation; groups of genus two [410] 303 The graphical representation of the simple group of order 168; deduction of its defining relations [419] 304—307 Cayley’s colour-groups [423] CHAPTER XX. ON CONGRUENCE GROUPS. 308 Object of the chapter: the homogeneous linear group [428] 309, 310 Its sub-group constituted by the operations of determinant unity; its self-conjugate operations [429] 311_313 Its self-conjugate sub-groups; its compositionfactors ; the simple group defined by it [431] 314 The case n=2; the fractional linear group [434] 315_320 The distribution of its operations in conjugate sets [436] 321—324 Tetrahedral, octahedral and icosahedral sub-groups contained in it [442] 325—327 Enumeration of all types of sub-groups contained in it [447] 328 Generalisation of the fractional linear group [451] Representation of the simple group defined by the linear homogeneous group as a doubly transitive permutation-group [452] Special cases of the linear homogeneous group; simple isomorphism between the alternating group of degree 8 and the group of isomorphisms of an Abelian group of order 16 and type (1, 1,1,1) [455] Generalisation of the homogeneous linear group; Examples [467] Note A. On the equation N = h1 + h2 + ...+hr [461] Note B. On the group of isomorphisms of a group [463] Note C. On the symmetric group [464] Note D. On the completely reduced form of a group of monomial substitutions [470] Note E. On tho irreducible representations of a group which has a self-conjugate sub-group of prime index [472] Note F. On groups of finite order which are simply isomorphic with irreducible groups of linear substitutions [476] Note G. On the representation of a group of finite order as a group of linear substitutions with rational coefficients [479] Note H. On the group of the twenty-seven lines on a cubic surface [485] Note I. On the conditions of reducibility of a group of linear substitutions of finite order [489] Note J. On conditions for the finiteness of the order of a group of linear substitutions [491] Note K. On the representation of a group of finite order as a group of birational transformations of an algebraic curve [496] Note L. On the group-characteristics of the fractional linear group [499] Note M. On groups of odd order [503] Note N. On the orders of simple groups [504] Note 0. On algebraic numbers [505] Index of TECHNICAL TERMS [507] Index of AUTHORS QUOTED [508] General index [609]
Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 20 B967 (Browse shelf) | Available | A-1896 |
Unabridged republication of the 2nd ed., published in 1911 [by Cambridge Univ. Press].
CONTENTS --
CHAPTER I. --
ON PERMUTATIONS. --
PAGE --
Object of the chapter [1] --
Definition of a permutation [1] --
Notation for permutations; cycles; products of permutations [1] --
Identical permutation; inverse permutations; order of a permutation [4] --
Circular, regular, similar and permutable permutations [7] --
Transpositions; representation of a permutation as a product of transpositions; odd and even permutations; Examples [9] --
CHAPTER II. --
THE DEFINITION OF A GROUP. --
Definition of a group [11] --
The identical operation [12] --
Continuous, mixed, and discontinuous groups [13] --
Order of an operation; product of operations; every operation of order mn, m and n relatively prime, can be expressed in just one way as the product of permutable operations of orders m and n [14] --
Examples of groups of operations; multiplication table of a group [17] --
Generating operations of a group; defining relations; simply isomorphic groups [20] --
Representation of a group of order N as a group of regular permutations of A symbols [22] --
Various modes of representing groups [24] --
CHAPTER III. --
ON THE SIMPLER PROPERTIES OF A GROUP WHICH ARE INDEPENDENT OF ITS MODE OF REPRESENTATION. --
Sub-groups; the order of a sub group is a factor of the order of the group containing it; various notations connected with a group and its sub-groups [25] --
Common sub-group of two groups; further notations. [27] --
Transforming one operation by another; conjugate operations and sub-groups; self-conjugate operations and self-con jugate sub-groups; Abelian groups; simple and composite groups [29] --
The operations of a group which are permutable with a given operation or sub-group form a group [31] --
Complete conjugate sets of operations and sub-groups [33] --
Theorems concerning self-conjugate sub-groups; maximum self-conjugate sub-groups; maximum subgroups [34] --
Multiply isomorphic groups; factor-groups; direct product of two groups [37] --
General isomorphism between two groups [41] --
Permutable groups; the group generated by two selfconjugate sub-groups of a given group; Examples [42] --
CHAPTER IV. --
FURTHER PROPERTIES OF A GROUP WHICH ARE INDEPENDENT --
OF ITS MODE OF REPRESENTATION. --
35 If pm(p prime) divides the order of a group, there is a sub-group of order pm [46] --
36 Groups of order p2 and pq [48] --
37 The number of operations of a group of order N whose nth powers are conjugate to a given operation is zero or a multiple of the highest common factor of N and n [49] --
38—40 Commutators; commutator sub-group or derived group; series of derived group; soluble groups; metabelian groups [54] --
41—43 Multiplication of conjugate sets; inverse sets; selfinverse set [57] --
44—47 Multiplication table of conjugate sets; deductions from it [60] --
CHAPTER V. --
ON THE COMPOSITION-SERIES OF A GROUP --
48 The composition-series, composition-factors and factorgroups of a given group [64] --
49, 50 Invariance of the factor-groups for different composition-series [65] --
51 Chief composition-series; invariance of its factor-groups [68] --
52, 53 Nature of the factor-groups of a chief-series; minimum self-conjugate sub-groups [69] --
54 Construction of a composition-series to contain a given chief-series [71] --
55, 56 Examples of composition-series [72] --
57, 58 Theorems concerning composition-series [74] --
59 Groups of order p2q [76] --
CHAPTER VI. --
ON THE ISOMORPHISM OF A GROUP WITH ITSELF. --
60 Object of the chapter [81] --
61 Definition of an isomorphism; identical isomorphism. [82] --
62 The group of isomorphisms of a group 82, [83] --
63 Inner and outer isomorphisms; the inner isomorphisms constitute a self-conjugate sub-group of the group of isomorphisms [84] --
64 The holomorph of a group [86] --
65, 66 Isomorphisms which permute the conjugate sets [88] --
67 Permutation of sub-groups by the group of isomorphisms [91] --
68 Definition of a characteristic sub-group; nature of a group with no characteristic sub-group. [92] --
69 Characteristic-series of a group [93] --
70 Definition of a complete group; a group with a complete group as a self-conjugate sub-group must be a direct product [93] --
71, 72 Theorems concerning complete groups [95] --
73 The orders of certain isomorphisms; Examples [97] --
CHAPTER VII. --
ON ABELIAN GROUPS. --
74 Introductory [99] --
75 Every Abelian group is the direct product of Abelian groups whose orders are powers of primes [100] --
76 Limitation of the discussion to Abelian groups whose orders are powers of primes [101] --
77 Existence of a set of independent generating operations for such a group [101] --
78 The orders of certain sub-groups of such a group [103] --
79, 80 Invariance of the orders of the generating operations ; simply isomorphic Abelian groups; symbol for Abelian group of given type [104] --
81 Determination of all types of sub-group of a given Abelian group [106] --
82 Characteristic series of an Abelian group [108] --
83, 84 Properties of an Abelian group of type (1, 1, ..., 1) [110] --
85 The group of isomorphisms and the holomorph of such a group [111] --
86 The orders of the isomorphisms of an Abelian group [112] --
87 The group of isomorphisms and the holomorph of any Abelian group [113] --
88 The group of isomorphisms and the holomorph of a cyclical group 114, [115] --
89, 90 The linear homogeneous group; Examples 116—119 --
CHAPTER VIII. --
ON GROUPS WHOSE ORDERS ARE THE POWERS OF PRIMES. --
91 Object of the chapter [119] --
92 Every group whose order is the power of a prime contains self-conjugate operations . [119] --
93 The series of self-conjugate sub-groups H1, H2, ..., Hn ,E, such that is the central of G|H i+1 [120] --
--
94, 95 The series of derived groups [120] --
96 Every sub-group is contained self-conjugately in a sub-group of greater order [122] --
97 The operations conjugate to a given operation [123] --
98, 99 Illustrations of preceding paragraphs [124] --
100 Operations conjugate to powers of themselves [126] --
101—103 Number of sub-groups of given order is congruent to 1, mod. p [128] --
104 Groups of order pm with a single sub-group of order p8 are cyclical, where p is an odd prime [130] --
105 Groups of order 2m with a single sub-group of order 28 are cyclical unless s is 1, in which case there is just one other type [131] --
106 The quaternion group [132] --
107 Some characteristic sub-groups [133] --
108, 109 Groups of order pm with a self-conjugate cyclical sub-group of order pm-1 [134] --
110, 111 Groups of order pm with a self-conjugate cyclical sub-group of order pm-2 [136] --
112 Distinct types of groups of orders p2 and p3 [139] --
113—116 Distinct types of groups of order p4 [140] --
117, 118 Tables of groups of orders p2, p3 and p4 [144] --
119 Examples [146] --
CHAPTER IX. --
ON SY LOW’S THEOREM. --
120 Proof of Sylow’s theorem [149] --
121 Generalisation of Sylow’s theorem [152] --
122 . Theorem concerning the maximum common subgroup of two Sylow sub-groups [153] --
123—125 Further theorems concerning Sylow sub-groups [154] --
126 Determination of all distinct types of group of order 24 [167] --
127 Determination of the only group of order 60 with no self-conjugate sub-group of order 5 [161] --
128, 129 Groups whose Sylow sub-groups are all cyclical; their defining relations [163] --
130 Groups with properties analogous to those of groups whose orders are powers of primes; Examples [166] --
CHAPTER X. --
ON PERMUTATION-GROUPS : TRANSITIVE AND INTRANSITIVE GROUPS: PRIMITIVE AND IMPRIMITIVE GROUPS. --
131 The degree of a permutation-group [168] --
132 The symmetric and the alternating groups . [169] --
133 Transitive and intransitive groups; the degree of --
a transitive group is a factor of the order . 170, [171] --
134 Transitive groups whose permutations, except identity, permute all or all but one of the symbols [171] --
135 Conjugate permutations are similar; self-conjugate operations and self-conjugate sub-groups of a transitive group [173] --
136 Transitive groups of which the order is equal to the degree [174] --
137 Multiply transitive groups; the order of a k-ply transitive group of degree n is divisible by re(n—1) ...(n —k+1) [176] --
138 Groups of degree n, which do not contain the alternating group, cannot be more than ((1/3)n+l)-ply transitive [178] --
139 The alternating group of degree re is simple except when n is 4 [180] --
140, 141 Examples of doubly and triply transitive groups . [181] --
142—144 Intransitive groups; transitive constituents; the general isomorphisms between two groups [186] --
145 Tests of transitivity [189] --
146 Definition of primitivity and imprimitivity; imprimitive systems [191] --
147 Test of primitivity [192] --
148 Properties of imprimitive systems .... [194] --
149 Self-conjugate sub-groups of transitive groups; a self-conjugate sub-group of a primitive group must be transitive [195] --
150 Self-conjugate sub-groups of ir-ply transitive groups are in general (k —l)-ply transitive [197] --
151—154 Further theorems concerning self-conjugate subgroups of multiply transitive groups; a group which is at least doubly transitive must, in general, either be simple or contain a simple group as a self-conjugate sub-group [198] --
155 Construction of a primitive group with an imprimitive self-conjugate sub-group . [202] --
156 Examples [203] --
CHAPTER XI. --
ON PERMUTAT1ON-GROUPS : TRAN8ITIVITY AND PR1MITIVITY: (concluding properties). --
157—160 Primitive groups with transitive sub-groups of smaller degree; limit to the order of a primitive group of given degree [205] --
161 Property of the symmetric group . . . . [208] --
162 The symmetric group of degree n is a complete group except when n is 6; the group of isomorphisms of the symmetric group of degree 6 [209] --
163—165 Further limitations on the order of a primitive group; examples of the same [210] --
166 Determination of all primitive groups whose degrees do not exceed 8 [214] --
167—169 Sub-groups of doubly transitive groups which leave two symbols unchanged; complete sets of triplets [221] --
170, 171 The most general permutation-group each of whose operations is permutable with a given permutation, or with every permutation of a given group [224] --
172 The most general transitive group whose order is the power of a prime [227] --
173 Examples [229] --
CHAPTER XII. --
ON THE REPRESENTATION OF A GROUP OF FINITE ORDER AS A PERMUTATION-GROUP. --
174 Definition of representation; equivalent and distinct representations [231] --
175 The two representations of a group as a regular permutation-group given by pre- and postmultiplication are equivalent [232] --
176 The imprimitive systems in the representation of a group as a regular permutation-group [232] --
177—179 To each conjugate set of sub-groups there corresponds a transitive representation; every transitive representation arises in this way [233] --
180, 181 The mark of a sub-group in a representation; table of marks; there are just s distinct representations, where s is the number of distinct conjugate sets of sub-groups [236] --
182, 183 The same set of permutations may give two or more distinct representations ; connection with outer isomorphisms [239] --
184, 185 Composition of representations; number of representations of given degree [240] --
186 A more general definition of equivalence [241] --
187 Alternative process for setting up representations [242] --
CHAPTER XIII. --
ON GROUPS OF LINEAR SUBSTITUTIONS J REDUCIBLE AND IRREDUCIBLE GROUPS. --
188, 189 Linear substitutions; their determinants; groups of linear substitutions [243] --
190 Transposed groups of linear substitutions; conjugate groups of linear substitutions; generalisation [245] --
191 Composition of isomorphic groups of linear substitutions [247] --
192 Characteristic equation of a substitution; characteristic of a substitution [249] --
193 Canonical form of a linear substitution of finite order [251] --
194 Definition of an Hermitian form; definite forms; properties of a definite form [253] --
195 Existence of a definite Hermitian form which is invariant for a group and its conjugate. [255] --
196 Standard form for a group of linear substitutions of finite order [256] --
197 Reducible and irreducible groups of linear substitutions ; completely reducible groups [258] --
198—200 A group of linear substitutions of finite order is either irreducible or completely reducible [259] --
201 Proof of the preceding result when the coefficients are limited to a given algebraic field [264] --
202 Substitutions permutable with every substitution of an irreducible group [265] --
203 The group of linear substitutions permutable with every substitution of a given group of linear substitutions; Examples; Note [266] --
CHAPTER XIV. --
ON THE REPRESENTATION OF A GROUP OF FINITE ORDER AS A GROUP OF LINEAR SUBSTITUTIONS. --
Definition of a representation; distinct and equivalent representations; Examples [269] --
The identical representation; irreducible components of a representation; reduced variables [271] --
The number of linearly independent invariant Hermitian forms for a representation and its conjugate [272] --
The completely reduced form of any representation of a group as a transitive permutation-group [273] --
The completely reduced form of the representation of a group as a regular permutation-group; all the irreducible representations occur in it; the number of distinct irreducible representations is equal to the number of conjugate sets [276] --
Irreducible representations with which the group is multiply isomorphic; irreducible representations in a single symbol [278] --
CHAPTER XV. --
ON GROUP-CHARACTERISTICS. --
211 Explanation of the notation [280] --
212 Set of group-characteristics; in conjugate representations corresponding group-characteristics are conjugate imaginaries [281] --
Proof of relations between the sets of group-characteristics [283] --
Two representations of a group are equivalent if, and only if, they have the same group-characteristics [287] --
Relations between the representation of a group as a transitive permutation-group when the more general definition of equivalence is used [288] --
Further relations between the group-character-istics; table of relations [290] --
Composition of the irreducible representations Two distinct conjugate sets cannot have the same characteristic in every representation [293] --
Case of groups of odd order [294] --
Determination of the characteristics from the multiplication table of the conjugate sets; Example [295] --
The number of variables operated on by an irreducible representation is a factor of the order of the group [297] --
Property of set of irreducible representations which combine among themselves by composition [298] --
Completely reduced form of the group on the homogeneous products of the variables operated on by a group of linear substitutions [300] --
The irreducible representation and conjugate sets of a factor-group [301] --
The reduction of a regular permutation-group ; the complete reduction of the general group {G, G'} of § 136 [302] --
The representation of the simple groups of orders 60 and 168 as irreducible groups in 3 variables [307] --
Nature of the coefficients in a group of linear substitutions of finite order [311] --
Families of irreducible representations; the number of families is equal to the number of distinct conjugate sets of cyclical sub-groups [311] --
The characteristics of a family of representations Invariant property of the multiplication table of conjugate sets [316] --
Similar invariant property of the composition table of the irreducible representations [.317] --
Examples [318] --
CHAPTER XVI. --
SOME APPLICATIONS OF THE THEORY OF GROUPS OF LINEAR --
SUBSTITUTIONS AND OF GROUP-CHARACTERISTICS. --
§§ PAGE --
239 Introductory [321] --
240, 241 Groups of order are soluble [321] --
242 Representation of a group as a group of monomial substitutions [324] --
243 Application of this representation to obtain conditions for the existence of self-conjugate sub-groups [325] --
244 Particular cases; a group whose order is not divisible by 12 or by the cube of a prime is soluble [327] --
245 Further particular cases; the order, if even, of a simple group is divisible by 12, 16 or 56 [328] --
246 Relations between the characteristics of a group and those of any sub-group [330] --
247 A transitive permutation-group whose operations permute all or all but one of the symbols has a regular self-conjugate sub-group [331] --
248 Groups of isomorphisms which leave E only unchanged [334] --
249 Isomorphisms which change each conjugate set into itself [336] --
250 The irreducible components of a transitive permutation-group [338] --
251 Simply transitive groups of prime degree are soluble [339] --
252 Generalisation of preceding theorem [341] --
253 On the result of compounding an irreducible group with itself; some properties of groups of odd order [343] --
254 Criterion for the existence of operations of composite order [346] --
255 On certain Abelian sub-groups of irreducible groups [348] --
256, 257 Congruences between characteristics which indicate the existence of self-conjugate sub-groups; illustrations [349] --
258 Every irreducible representation of a group whose order is the power of a prime can be expressed as a group of monomial substitutions [351] --
259 Examples [353] --
CHAPTER XVII. --
ON THE INVARIANTS OF GROUPS OF LINEAR SUBSTITUTIONS. --
260, 261 Definition of invariants and relative invariants; condition for existence of relative invariants; invariant in the form of a rational fraction [355] --
262 Existence of an algebraically independent set of invariants [357] --
263 Formation, for a group in n variables, of a set of n+1 invariants in terms of which all invariants are rationally expressible [357] --
264 On the possibility of replacing the above set of n+1 invariants by a set of n [360] --
265 The group of linear substitutions for which each of a given set of functions is invariant [360] --
266—268 Examples of sets of invariants for certain special groups [362] --
269 Property of invariants of an irreducible group [366] --
270 Condition that an irreducible group may have a quadratic invariant [367] --
271 General remarks on the relation of a group to its invariants [369] --
272 Examples [370] --
CHAPTER XVIII. --
ON THE GRAPHICAL REPRESENTATION OF A GROUP. --
Introductory remarks [372] --
The most general discontinuous group that can be generated by a finite number of operations ; the relation of this group to the special group that arises when one or more relations hold between the generating operations [373] --
Graphical representation of a cyclical group [376] --
Graphical representation of the general group, when no relations connect the generating operations [379] --
Graphical representation of the special group when relations connect the generating operations [384] --
Illustration of the preceding paragraphs [386] --
Graphical representation of the special group when the generating operations are of finite order, [389] --
287, 288 Graphical representation of a group of finite order [394] --
289 The genus of a group [397] --
290, 291 Limitation on the order and on the number of defining relations of a group of given genus: Examples; Note [398] --
CHAPTER XIX. --
ON THE GRAPHICAL REPRESENTATION OF GROUPS: GROUPS OF GENUS ZERO AND UNITY: CAYLEY’S COLOUR-GROUPS. --
292 The diophantine relation connecting the order, the genus, and the number and orders of the generating operations [402] --
293—296 Groups of genus zero : their defining relations and graphical representation; the dihedral, tetrahedral, octahedral and icosahedral groups 403—409 297—302 Groups of genus unity: their defining relations and graphical representation; groups of genus two [410] --
303 The graphical representation of the simple group of order 168; deduction of its defining relations [419] --
304—307 Cayley’s colour-groups [423] --
CHAPTER XX. --
ON CONGRUENCE GROUPS. --
308 Object of the chapter: the homogeneous linear group [428] --
309, 310 Its sub-group constituted by the operations of determinant unity; its self-conjugate operations [429] --
311_313 Its self-conjugate sub-groups; its compositionfactors ; the simple group defined by it [431] --
314 The case n=2; the fractional linear group [434] --
315_320 The distribution of its operations in conjugate sets [436] --
321—324 Tetrahedral, octahedral and icosahedral sub-groups contained in it [442] --
325—327 Enumeration of all types of sub-groups contained in it [447] --
328 Generalisation of the fractional linear group [451] --
Representation of the simple group defined by the linear homogeneous group as a doubly transitive permutation-group [452] --
Special cases of the linear homogeneous group; simple isomorphism between the alternating group of degree 8 and the group of isomorphisms of an Abelian group of order 16 and type (1, 1,1,1) [455] --
Generalisation of the homogeneous linear group; Examples [467] --
Note A. On the equation N = h1 + h2 + ...+hr [461] --
Note B. On the group of isomorphisms of a group [463] --
Note C. On the symmetric group [464] --
Note D. On the completely reduced form of a group of monomial substitutions [470] --
Note E. On tho irreducible representations of a group which has a self-conjugate sub-group of prime index [472] --
Note F. On groups of finite order which are simply isomorphic with irreducible groups of linear substitutions [476] --
Note G. On the representation of a group of finite order as a group of linear substitutions with rational coefficients [479] --
Note H. On the group of the twenty-seven lines on a cubic surface [485] --
Note I. On the conditions of reducibility of a group of linear substitutions of finite order [489] --
Note J. On conditions for the finiteness of the order of a group of linear substitutions [491] --
Note K. On the representation of a group of finite order as a group of birational transformations of an algebraic curve [496] --
Note L. On the group-characteristics of the fractional linear group [499] --
Note M. On groups of odd order [503] --
Note N. On the orders of simple groups [504] --
Note 0. On algebraic numbers [505] --
Index of TECHNICAL TERMS [507] --
Index of AUTHORS QUOTED [508] --
General index [609] --
MR, 16,1086c
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