Introduction to the theory of groups of finite order / by Robert D. Carmichael.
Editor: [New York] : Dover, 1956, c1937Descripción: xiv, 447 p. ; 21 cmOtra clasificación: 20-01CHAPTER I • INTRODUCTION
1. Sets, Systems, and Groups [3]
2. Permutations [5]
3. Definition of Group [15]
4. Certain Permutation Groups [19]
5. Properties of the Elements of a Group [26]
6. Subgroups [28]
7. Some Classes of Groups [29]
8. Generators of Groups [30]
9. Simple Isomorphism. Abstract Groups [31]
Miscellaneous Exercises [39]
CHAPTER II • FIVE FUNDAMENTAL THEOREMS
10. Orders of Subgroups [44]
11. Miscellaneous Theorems. Conjugate Elements and Subgroups [45]
12. Representation of an Abstract Finite Group as a Regular Permutation Group [53]
13. Sylow’s Theorem [58]
14. Generators of Abelian Groups [66]
15. Prime-Power Groups [67]
Miscellaneous Exercises [71]
CHAPTER III • ADDITIONAL PROPERTIES OF GROUPS IN GENERAL
16. Isomorphism [74]
17. Isomorphisms of a Group with Itself [76]
18. The Holomorph of a Group [79]
19. On Certain Subgroups of a Group G [81]
20. Factor-Groups [84]
21. The Composition-Series of a Group [88]
22. The Theorem of Frobenius [92]
Miscellaneous Exercises [95]
CHAPTER IV • ABELIAN GROUPS
23. Classification of Abelian Groups [98]
24. Abelian Groups of a Given Order [101]
25. Subgroups of a Prime-Power Abelian Group [102]
26. Number of Elements of a Given Order [103]
27. Groups of Isomorphisms of Cyclic Groups [104]
28. Properties of an Abelian Group G of Order pm and Type (1, 1, • • •, 1) [107]
29. Analytical Representations of G, I, and K [108]
30. Groups of Isomorphisms of Abelian Groups in General [112]
31. Hamiltonian Groups [113]
Miscellaneous Exercises [118]
CHAPTER V • PRIME-POWER GROUPS
32. General Properties [120]
33. Some Self-conjugate Subgroups [122]
34. Number of Subgroups of Index p [123]
35. Number of Subgroups of Any Given Order [124]
36. Prime-Power Groups Each with a Single Subgroup of a Given Order [128]
37. Groups of Order pm Each with a Cyclic Subgroup of Index p [132]
Miscellaneous Exercises [136]
CHAPTER VI • PERMUTATION GROUPS
38. Introduction [138]
39. Transitive Groups [139]
40. Examples of Multiply Transitive Groups [143]
41. An Upper Limit to the Degree of Transitivity [148]
42. Simplicity of the Alternating Group of Degree n ≠ 4 [153]
43. Self-conjugate Subgroups of Symmetric Groups [154]
44. Representation of a Group as a Transitive Group [155]
45. Intransitive Groups [158]
46. Primitive and Imprimitive Groups [159]
Miscellaneous Exercises [163]
CHAPTER VII • DEFINING RELATIONS FOR ABSTRACT GROUPS
47. Introduction. Two General Theorems [166]
48. Symmetric and Alternating Groups [169]
49. Finite Groups {s, t} such that s2 = t2 [177]
50. Dihedral and Dicyclic Groups [181]
Miscellaneous Exercises [185]
CHAPTER VIII • GROUPS OF LINEAR TRANSFORMATIONS
51. Properties of Linear Substitutions [188]
52. Finite Groups of Linear Transformations [194]
53. Reducible and Irreducible Groups [200]
54. Composition of Isomorphic Groups [204]
55. Representation of a Finite Group as a Group of Linear Homogeneous Transformations [206]
56. Group Characteristics [210]
57. Regular Permutation Groups [217]
58. Certain Composite Groups [226]
59. Transitive Groups in Which Only the Identity Leaves Two Symbols Fixed [229]
60. Simply Transitive Groups of Prime Degree [234]
Miscellaneous Exercises [240]
CHAPTER IX • GALOIS FIELDS
61. Introduction [242]
62. Finite Fields [242]
63. Galois Fields [251]
64. Existence of Galois Fields [256]
65. Inclusion of One Finite Field within Another [260]
66. Analytical Representation of Permutations [263]
67. Linear Groups in One Variable in GF[pn] [265]
68. Linear Fractional Groups in One Variable in GF[pn] [266]
69. Certain Doubly Transitive Groups of Degree pn [267]
70. Certain Doubly and Triply Transitive Groups of Degree pn + 1 [272]
71. A Class of Simple Groups [276]
Miscellaneous Exercises [286]
CHAPTER X • GROUPS OF ISOMORPHISMS OF ABELIAN GROUPS OF ORDER pm AND TYPE (1, 1, • • •, 1)
72. Analytical Representation of Elements and Subgroups [289]
73. The General Linear Homogeneous Group GLH(k +1, pn) [291]
74. Analytical Representations of the Group I of Isomorphisms of the Group G(k+1)n with Itself [295]
75. On Certain Subgroups of I [300]
76. The Holomorph of G [303]
77. DoubLy Transitive Groups of Degree pn and Order pn(pn — 1) [306]
78. Analytical Forms of M [309]
79. On Certain Elements and Subgroups of M [310]
80. The Case of Certain Invariant Subgroups of M [313]
Miscellaneous Exercises [321]
CHAPTER XI • FINITE GEOMETRIES
81. Definition of the Finite Projective Geometries [323]
82. Representation of Finite Geometries by Means of Galois Fields [324]
83. Representation of Finite Geometries by Means of Abelian Groups [328]
84. Euclidean Finite Geometries EG(k, pn) [329]
85. The Principle of Duality [330]
86. Finite Geometries Contained within Finite Geometries [334]
87. Interrelations of Finite Geometries and Abelian Groups [337]
88. Some Generalizations [342]
89. Geometric Sets of Subgroups [344]
90. Another Analytical Representation of PG(k, pn) [345]
91. Configurations in PG (K, pn) [346]
Miscellaneous Exercises [352]
CHAPTER XII • COLLINEATION GROUPS IN THE FINITE GEOMETRIES
92. The Projective Group in PG(k, pn) [355]
93. The Collineation Group in PG (k, pn) [358]
94. Subgroups of the Collineation Group [362]
95. Collineation Groups Leaving an EG(k, pn) Invariant [374]
96. Collineation Groups Leaving Other Subspaces Invariant [383]
97. Some Special Cases [385]
98. Generators of Certain Multiply Transitive Groups of Prime Degree [387]
99. Groups of Certain Prime Degrees [389]
Miscellaneous Exercises [393]
CHAPTER XIII • ALGEBRAS OF DOUBLY TRANSITIVE GROUPS OF DEGREE pn AND ORDER p”(pn - 1)
100. On the Definition of Group [395]
101. Definition of Algebras A[s] [396]
102. Construction of Algebras A[s] by Means of Certain Doubly Transitive Groups [398]
103. Linear Transformations on the Marks of an A[s] [400]
104. Simple Isomorphism of Algebras A[pn] [402]
105. Integral Elements of an Algebra A[pn] [404]
106. Analytical Representations of Algebras A[pn] [405]
107. The Algebras A σ,1[pn] [406]
108. Analytic Finite Plane Geometries [407]
Miscellaneous Exercises [413]
CHAPTER XIV • TACTICAL CONFIGURATIONS
109. Immediate Examples [415]
110. Configurations Associated with Coble’s Box Porism [417]
111. Certain Additional Configurations [418]
112. Subgeometries and the Complementary Sets [419]
113. Triple Systems and Triple Groups [425]
114. Quadruple Systems [429]
115. Configurations Associated with the Mathieu Groups [431]
116. Some Generalizations [433]
117. Certain Complete 2-2-k-Configurations [435]
Miscellaneous Exercises [438]
INDEX [443]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 20 C287 (Browse shelf) | Available | A-710 |
Unabridged and unaltered republication of the 1st ed.
CHAPTER I • INTRODUCTION --
1. Sets, Systems, and Groups [3] --
2. Permutations [5] --
3. Definition of Group [15] --
4. Certain Permutation Groups [19] --
5. Properties of the Elements of a Group [26] --
6. Subgroups [28] --
7. Some Classes of Groups [29] --
8. Generators of Groups [30] --
9. Simple Isomorphism. Abstract Groups [31] --
Miscellaneous Exercises [39] --
CHAPTER II • FIVE FUNDAMENTAL THEOREMS --
10. Orders of Subgroups [44] --
11. Miscellaneous Theorems. Conjugate Elements and Subgroups [45] --
12. Representation of an Abstract Finite Group as a Regular Permutation Group [53] --
13. Sylow’s Theorem [58] --
14. Generators of Abelian Groups [66] --
15. Prime-Power Groups [67] --
Miscellaneous Exercises [71] --
CHAPTER III • ADDITIONAL PROPERTIES OF GROUPS IN GENERAL --
16. Isomorphism [74] --
17. Isomorphisms of a Group with Itself [76] --
18. The Holomorph of a Group [79] --
19. On Certain Subgroups of a Group G [81] --
20. Factor-Groups [84] --
21. The Composition-Series of a Group [88] --
22. The Theorem of Frobenius [92] --
Miscellaneous Exercises [95] --
CHAPTER IV • ABELIAN GROUPS --
23. Classification of Abelian Groups [98] --
24. Abelian Groups of a Given Order [101] --
25. Subgroups of a Prime-Power Abelian Group [102] --
26. Number of Elements of a Given Order [103] --
27. Groups of Isomorphisms of Cyclic Groups [104] --
28. Properties of an Abelian Group G of Order pm and Type (1, 1, • • •, 1) [107] --
29. Analytical Representations of G, I, and K [108] --
30. Groups of Isomorphisms of Abelian Groups in General [112] --
31. Hamiltonian Groups [113] --
Miscellaneous Exercises [118] --
CHAPTER V • PRIME-POWER GROUPS --
32. General Properties [120] --
33. Some Self-conjugate Subgroups [122] --
34. Number of Subgroups of Index p [123] --
35. Number of Subgroups of Any Given Order [124] --
36. Prime-Power Groups Each with a Single Subgroup of a Given Order [128] --
37. Groups of Order pm Each with a Cyclic Subgroup of Index p [132] --
Miscellaneous Exercises [136] --
CHAPTER VI • PERMUTATION GROUPS --
38. Introduction [138] --
39. Transitive Groups [139] --
40. Examples of Multiply Transitive Groups [143] --
41. An Upper Limit to the Degree of Transitivity [148] --
42. Simplicity of the Alternating Group of Degree n ≠ 4 [153] --
43. Self-conjugate Subgroups of Symmetric Groups [154] --
44. Representation of a Group as a Transitive Group [155] --
45. Intransitive Groups [158] --
46. Primitive and Imprimitive Groups [159] --
Miscellaneous Exercises [163] --
CHAPTER VII • DEFINING RELATIONS FOR ABSTRACT GROUPS --
47. Introduction. Two General Theorems [166] --
48. Symmetric and Alternating Groups [169] --
49. Finite Groups {s, t} such that s2 = t2 [177] --
50. Dihedral and Dicyclic Groups [181] --
Miscellaneous Exercises [185] --
CHAPTER VIII • GROUPS OF LINEAR TRANSFORMATIONS --
51. Properties of Linear Substitutions [188] --
52. Finite Groups of Linear Transformations [194] --
53. Reducible and Irreducible Groups [200] --
54. Composition of Isomorphic Groups [204] --
55. Representation of a Finite Group as a Group of Linear Homogeneous Transformations [206] --
56. Group Characteristics [210] --
57. Regular Permutation Groups [217] --
58. Certain Composite Groups [226] --
59. Transitive Groups in Which Only the Identity Leaves Two Symbols Fixed [229] --
60. Simply Transitive Groups of Prime Degree [234] --
Miscellaneous Exercises [240] --
CHAPTER IX • GALOIS FIELDS --
61. Introduction [242] --
62. Finite Fields [242] --
63. Galois Fields [251] --
64. Existence of Galois Fields [256] --
65. Inclusion of One Finite Field within Another [260] --
66. Analytical Representation of Permutations [263] --
67. Linear Groups in One Variable in GF[pn] [265] --
68. Linear Fractional Groups in One Variable in GF[pn] [266] --
69. Certain Doubly Transitive Groups of Degree pn [267] --
70. Certain Doubly and Triply Transitive Groups of Degree pn + 1 [272] --
71. A Class of Simple Groups [276] --
Miscellaneous Exercises [286] --
CHAPTER X • GROUPS OF ISOMORPHISMS OF ABELIAN GROUPS OF ORDER pm AND TYPE (1, 1, • • •, 1) --
72. Analytical Representation of Elements and Subgroups [289] --
73. The General Linear Homogeneous Group GLH(k +1, pn) [291] --
74. Analytical Representations of the Group I of Isomorphisms of the Group G(k+1)n with Itself [295] --
75. On Certain Subgroups of I [300] --
76. The Holomorph of G [303] --
77. DoubLy Transitive Groups of Degree pn and Order pn(pn — 1) [306] --
78. Analytical Forms of M [309] --
79. On Certain Elements and Subgroups of M [310] --
80. The Case of Certain Invariant Subgroups of M [313] --
Miscellaneous Exercises [321] --
CHAPTER XI • FINITE GEOMETRIES --
81. Definition of the Finite Projective Geometries [323] --
82. Representation of Finite Geometries by Means of Galois Fields [324] --
83. Representation of Finite Geometries by Means of Abelian Groups [328] --
84. Euclidean Finite Geometries EG(k, pn) [329] --
85. The Principle of Duality [330] --
86. Finite Geometries Contained within Finite Geometries [334] --
87. Interrelations of Finite Geometries and Abelian Groups [337] --
88. Some Generalizations [342] --
89. Geometric Sets of Subgroups [344] --
90. Another Analytical Representation of PG(k, pn) [345] --
91. Configurations in PG (K, pn) [346] --
Miscellaneous Exercises [352] --
CHAPTER XII • COLLINEATION GROUPS IN THE FINITE GEOMETRIES --
92. The Projective Group in PG(k, pn) [355] --
93. The Collineation Group in PG (k, pn) [358] --
94. Subgroups of the Collineation Group [362] --
95. Collineation Groups Leaving an EG(k, pn) Invariant [374] --
96. Collineation Groups Leaving Other Subspaces Invariant [383] --
97. Some Special Cases [385] --
98. Generators of Certain Multiply Transitive Groups of Prime Degree [387] --
99. Groups of Certain Prime Degrees [389] --
Miscellaneous Exercises [393] --
CHAPTER XIII • ALGEBRAS OF DOUBLY TRANSITIVE GROUPS OF DEGREE pn AND ORDER p”(pn - 1) --
100. On the Definition of Group [395] --
101. Definition of Algebras A[s] [396] --
102. Construction of Algebras A[s] by Means of Certain Doubly Transitive Groups [398] --
103. Linear Transformations on the Marks of an A[s] [400] --
104. Simple Isomorphism of Algebras A[pn] [402] --
105. Integral Elements of an Algebra A[pn] [404] --
106. Analytical Representations of Algebras A[pn] [405] --
107. The Algebras A σ,1[pn] [406] --
108. Analytic Finite Plane Geometries [407] --
Miscellaneous Exercises [413] --
CHAPTER XIV • TACTICAL CONFIGURATIONS --
109. Immediate Examples [415] --
110. Configurations Associated with Coble’s Box Porism [417] --
111. Certain Additional Configurations [418] --
112. Subgeometries and the Complementary Sets [419] --
113. Triple Systems and Triple Groups [425] --
114. Quadruple Systems [429] --
115. Configurations Associated with the Mathieu Groups [431] --
116. Some Generalizations [433] --
117. Certain Complete 2-2-k-Configurations [435] --
Miscellaneous Exercises [438] --
INDEX [443] --
MR, 17,823a
Click on an image to view it in the image viewer
There are no comments on this title.