The theory of groups / Marshall Hall, Jr.
Editor: New York : Macmillan, c1959Descripción: xiii, 434 p. ; 22 cmOtra clasificación: 20-021. INTRODUCTION [1] 1.1 Algebraic Laws [1] 1.2 Mappings [2] 1.3 Definitions for Groups and Some Related Systems [4] 1.4 Subgroups, Isomorphisms, Homomorphisms [7] 1_5 Cosets. Theorem of Lagrange. Cyclic Groups. Indices [10] 1.6 Conjugates and Classes [13] 1.7 Double Cosets [14] 1.8 Remarks on Infinite Groups [15] 1.9 Examples of Groups [19] 2. NORMAL SUBGROUPS AND HOMOMORPHISMS [26] 2.1 Normal Subgroups [26] 2.2 The Kernel of a Homomorphism [27] 2.3 Factor Groups [27] 2.4 Operators [29] 2.5 Direct Products and Cartesian Products [32] 3. ELEMENTARY THEORY OF ABELIAN GROUPS [35] 3.1 Definition of Abelian Group. Cyclic Groups [35] 3.2 Some Structure Theorems for Abelian Groups [36] 3.3 Finite Abelian Groups. Invariants [40] 4. SYLOW THEOREMS [43] 4.1 Falsity of the Converse of the Theorem of Lagrange [43] 4.2 The Three Sylow Theorems [44] 4.3 Finite p-Groups [47] 4.4 Groups of Orders p, p2, pq, p3 [49] 5. PERMUTATION GROUPS [53] 5.1 Cycles [53] 5.2 Transitivity [55] 5.3 Representations of a Group by Permutations [56] 5.4 The Alternating Group A [59] 5.5 Intransitive Groups. Subdirect Products [63] 5.6 Primitive Groups [64] 5.7 Multiply Transitive Groups [68] 5.8 On a Theorem of Jordan [72] 5.9 The Wreath Product. Sylow Subgroups of Symmetric Groups [81] 6. AUTOMORPHISMS [84] 6.1 Automorphisms of Algebraic Systems [84] 6.2 Automorphisms of Groups. Inner Automorphisms [84] 6.3 The Holomorph of a Group [86] 6.4 Complete Groups [87] 6.5 Normal or Semi-direct Products [88] 7. FREE GROUPS [91] 7.1 Definition of Free Group [91] 7.2 Subgroups of Free Groups. The Schreier Method [94] 7.3 Free Generators of Subgroups of Free Groups. The Nielsen Method [106] 8. LATTICES AND COMPOSITION SERIES [115] 8.1 Partially Ordered Sets [115] 8.2 Lattices [116] 8.3 Modular and Semi-modular Lattices [117] 8.4 Principal Series and Composition Series [123] 8.5 Direct Decompositions [127] 8.6 Composition Series in Groups [131] 9. A THEOREM OF FROBENIUS; SOLVABLE GROUPS [136] 9.1 A Theorem of Frobenius [136] 9.2 Solvable Groups [138] 9.3 Extended Sylow Theorems in Solvable Groups [141] 9.4 Further Results on Solvable Groups [145] 10. SUPERSOLVABLE AND NILPOTENT GROUPS [149] 10.1 Definitions [149] 10.2 The Lower and Upper Central Series [149] 10.3 Theory of Nilpotent Groups [153] 10.4 The Frattini Subgroup of a Group [156] 10.5 Supersolvable Groups [158] 11. BASIC COMMUTATORS [165] 11.1 The Collecting Process [165] 11.2 The Witt Formulae. The Basis Theorem [168] 12. THE THEORY OF p-GROUPS; REGULAR p-GROUPS [176] 12.1 Elementary results [176] 12.2 The Bumside Basis Theorem. Automorphisms of p-Groups [176] 12.3 The Collection Formula [178] 12.4 Regular p-Groups [183] 12.5 Some Special p-Groups. Hamiltonian Groups [187] 13. FURTHER THEORY OF ABELIAN GROUPS [193] 13.1 Additive Groups. Groups Modulo One [193] 13.2 Characters of Abelian Groups. Duality of Abelian Groups [194] 13.3 Divisible Groups [197] 13.4 Pure Subgroups [198] 13.5 General Remarks [199] 14. MONOMIAL REPRESENTATIONS AND THE TRANSFER [200] 14.1 Monomial Permutations [200] 14.2 The Transfer [201] 14.3 A Theorem of Bumside [203] 14.4 Theorems of P. Hall, Grün, and Wielandt [204] 15. GROUP EXTENSIONS AND COHOMOLOGY OF GROUPS [218] 15.1 Composition of Normal Subgroup and Factor Group [218] 15.2 Central Extensions [222] 15.3 Cyclic Extensions [224] 15.4 Defining Relations and Extensions [226] 15.5 Group Rings and Central Extensions [228] 15.6 Double Modules [235] 15.7 Cochains, Coboundaries, and Cohomology Groups [236] 15.8 Applications of Cohomology to Extension Theory [240] 16. GROUP REPRESENTATION [247] 16.1 General Remarks [247] 16.2 Matrix Representation. Characters [247] 16.3 The Theorem of Complete Reducibility [251] 16.4 Semi-simple Group Rings and Ordinary Representations [255] 16.5 Absolutely Irreducible Representations. Structure of Simple Rings [262] 16.6 Relations on Ordinary Characters [267] 16.7 Imprimitive Representations [281] 16.8 Some Applications of the Theory of Characters [285] 16.9 Unitary and Orthogonal Representations [294] 16.10 Some Examples of Group Representation [298] 17. FREE AND AMALGAMATED PRODUCTS [311] 17.1 Definition of Free Product [311] 17.2 Amalgamated Products [312] 17.3 The Theorem of Kurosch [315] 18. THE BURNSIDE PROBLEM [320] 18.1 Statement of the Problem [320] 18.2 The Bumside Problem for n=2 and n=3 [320] 18.3 Finiteness of B(4,r) [324] 18.4 The Restricted Bumside problem. Theorems of P. Hall and G. Higman. Finiteness of B(6,r) [325] 19. LATTICES OF SUBGROUPS [339] 19.1 General Properties [339] 19.2 Locally Cyclic Groups and Distributive Lattices [340] 19.3 The Theorem of Iwasawa [342] 20. GROUP THEORY AND PROJECTIVE PLANES [346] 20.1 Axioms [346] 20.2 Collineations and the Theorem of Desargues [348] 20.3 Introduction of Coordinates [353] 20.4 Veblen-Wedderbum Systems. Hall Systems [356] 20.5 Moufang and Desarguesian Planes [366] 20.6 The Theorem of Wedderburn and the Artin-Zom Theorem [375] 20.7 Doubly Transitive Groups and Near-Fields [382] 20.8 Finite Planes. The Bruck-Ryser Theorem [392] 20.9 Collineations in Finite Planes [398] BIBLIOGRAPHY [421] INDEX [429] INDEX OF SPECIAL SYMBOLS [433]
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1. INTRODUCTION [1] --
1.1 Algebraic Laws [1] --
1.2 Mappings [2] --
1.3 Definitions for Groups and Some Related Systems [4] --
1.4 Subgroups, Isomorphisms, Homomorphisms [7] --
1_5 Cosets. Theorem of Lagrange. Cyclic Groups. Indices [10] --
1.6 Conjugates and Classes [13] --
1.7 Double Cosets [14] --
1.8 Remarks on Infinite Groups [15] --
1.9 Examples of Groups [19] --
2. NORMAL SUBGROUPS AND HOMOMORPHISMS [26] --
2.1 Normal Subgroups [26] --
2.2 The Kernel of a Homomorphism [27] --
2.3 Factor Groups [27] --
2.4 Operators [29] --
2.5 Direct Products and Cartesian Products [32] --
3. ELEMENTARY THEORY OF ABELIAN GROUPS [35] --
3.1 Definition of Abelian Group. Cyclic Groups [35] --
3.2 Some Structure Theorems for Abelian Groups [36] --
3.3 Finite Abelian Groups. Invariants [40] --
4. SYLOW THEOREMS [43] --
4.1 Falsity of the Converse of the Theorem of Lagrange [43] --
4.2 The Three Sylow Theorems [44] --
4.3 Finite p-Groups [47] --
4.4 Groups of Orders p, p2, pq, p3 [49] --
5. PERMUTATION GROUPS [53] --
5.1 Cycles [53] --
5.2 Transitivity [55] --
5.3 Representations of a Group by Permutations [56] --
5.4 The Alternating Group A [59] --
5.5 Intransitive Groups. Subdirect Products [63] --
5.6 Primitive Groups [64] --
5.7 Multiply Transitive Groups [68] --
5.8 On a Theorem of Jordan [72] --
5.9 The Wreath Product. Sylow Subgroups of Symmetric Groups [81] --
6. AUTOMORPHISMS [84] --
6.1 Automorphisms of Algebraic Systems [84] --
6.2 Automorphisms of Groups. Inner Automorphisms [84] --
6.3 The Holomorph of a Group [86] --
6.4 Complete Groups [87] --
6.5 Normal or Semi-direct Products [88] --
7. FREE GROUPS [91] --
7.1 Definition of Free Group [91] --
7.2 Subgroups of Free Groups. The Schreier Method [94] --
7.3 Free Generators of Subgroups of Free Groups. The Nielsen Method [106] --
8. LATTICES AND COMPOSITION SERIES [115] --
8.1 Partially Ordered Sets [115] --
8.2 Lattices [116] --
8.3 Modular and Semi-modular Lattices [117] --
8.4 Principal Series and Composition Series [123] --
8.5 Direct Decompositions [127] --
8.6 Composition Series in Groups [131] --
9. A THEOREM OF FROBENIUS; SOLVABLE GROUPS [136] --
9.1 A Theorem of Frobenius [136] --
9.2 Solvable Groups [138] --
9.3 Extended Sylow Theorems in Solvable Groups [141] --
9.4 Further Results on Solvable Groups [145] --
10. SUPERSOLVABLE AND NILPOTENT GROUPS [149] --
10.1 Definitions [149] --
10.2 The Lower and Upper Central Series [149] --
10.3 Theory of Nilpotent Groups [153] --
10.4 The Frattini Subgroup of a Group [156] --
10.5 Supersolvable Groups [158] --
11. BASIC COMMUTATORS [165] --
11.1 The Collecting Process [165] --
11.2 The Witt Formulae. The Basis Theorem [168] --
12. THE THEORY OF p-GROUPS; REGULAR p-GROUPS [176] --
12.1 Elementary results [176] --
12.2 The Bumside Basis Theorem. Automorphisms of p-Groups [176] --
12.3 The Collection Formula [178] --
12.4 Regular p-Groups [183] --
12.5 Some Special p-Groups. Hamiltonian Groups [187] --
13. FURTHER THEORY OF ABELIAN GROUPS [193] --
13.1 Additive Groups. Groups Modulo One [193] --
13.2 Characters of Abelian Groups. Duality of Abelian Groups [194] --
13.3 Divisible Groups [197] --
13.4 Pure Subgroups [198] --
13.5 General Remarks [199] --
14. MONOMIAL REPRESENTATIONS AND THE TRANSFER [200] --
14.1 Monomial Permutations [200] --
14.2 The Transfer [201] --
14.3 A Theorem of Bumside [203] --
14.4 Theorems of P. Hall, Grün, and Wielandt [204] --
15. GROUP EXTENSIONS AND COHOMOLOGY OF GROUPS [218] --
15.1 Composition of Normal Subgroup and Factor Group [218] --
15.2 Central Extensions [222] --
15.3 Cyclic Extensions [224] --
15.4 Defining Relations and Extensions [226] --
15.5 Group Rings and Central Extensions [228] --
15.6 Double Modules [235] --
15.7 Cochains, Coboundaries, and Cohomology Groups [236] --
15.8 Applications of Cohomology to Extension Theory [240] --
16. GROUP REPRESENTATION [247] --
16.1 General Remarks [247] --
16.2 Matrix Representation. Characters [247] --
16.3 The Theorem of Complete Reducibility [251] --
16.4 Semi-simple Group Rings and Ordinary Representations [255] --
16.5 Absolutely Irreducible Representations. Structure of Simple Rings [262] --
16.6 Relations on Ordinary Characters [267] --
16.7 Imprimitive Representations [281] --
16.8 Some Applications of the Theory of Characters [285] --
16.9 Unitary and Orthogonal Representations [294] --
16.10 Some Examples of Group Representation [298] --
17. FREE AND AMALGAMATED PRODUCTS [311] --
17.1 Definition of Free Product [311] --
17.2 Amalgamated Products [312] --
17.3 The Theorem of Kurosch [315] --
18. THE BURNSIDE PROBLEM [320] --
18.1 Statement of the Problem [320] --
18.2 The Bumside Problem for n=2 and n=3 [320] --
18.3 Finiteness of B(4,r) [324] --
18.4 The Restricted Bumside problem. Theorems of P. Hall and G. Higman. Finiteness of B(6,r) [325] --
19. LATTICES OF SUBGROUPS [339] --
19.1 General Properties [339] --
19.2 Locally Cyclic Groups and Distributive Lattices [340] --
19.3 The Theorem of Iwasawa [342] --
20. GROUP THEORY AND PROJECTIVE PLANES [346] --
20.1 Axioms [346] --
20.2 Collineations and the Theorem of Desargues [348] --
20.3 Introduction of Coordinates [353] --
20.4 Veblen-Wedderbum Systems. Hall Systems [356] --
20.5 Moufang and Desarguesian Planes [366] --
20.6 The Theorem of Wedderburn and the Artin-Zom --
Theorem [375] --
20.7 Doubly Transitive Groups and Near-Fields [382] --
20.8 Finite Planes. The Bruck-Ryser Theorem [392] --
20.9 Collineations in Finite Planes [398] --
BIBLIOGRAPHY [421] --
INDEX [429] --
INDEX OF SPECIAL SYMBOLS [433] --
MR, 21 #1996
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