Theory of groups of finite order / by W. Burnside.

Por: Burnside, William Snow, 1852-1927Editor: [New York?] : Dover, 1955Edición: 2nd edDescripción: xxiv, 512 p. ; 21 cmOtra clasificación: 20
Contenidos:
 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]
    Average rating: 0.0 (0 votes)
Item type Home library Shelving location Call number Materials specified Status Date due Barcode Course reserves
Libros Libros Instituto de Matemática, CONICET-UNS
Libros ordenados por tema 20 B967 (Browse shelf) Available A-1896

COMPLEMENTOS DE ÁLGEBRA


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

There are no comments on this title.

to post a comment.

Click on an image to view it in the image viewer

¿Necesita ayuda?

Si necesita ayuda para encontrar información, puede visitar personalmente la biblioteca en Av. Alem 1253 Bahía Blanca, llamarnos por teléfono al 291 459 5116, o enviarnos un mensaje a biblioteca.antonio.monteiro@gmail.com

Powered by Koha