Gorenstein D., Lyons R., Solomon R. — Classification of the Finite Simple Groups (Vol. 1) :: Электронная библиотека попечительского совета мехмата МГУ

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Gorenstein D., Lyons R., Solomon R. — Classification of the Finite Simple Groups (Vol. 1)

Gorenstein D., Lyons R., Solomon R. — Classification of the Finite Simple Groups (Vol. 1)

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Название: Classification of the Finite Simple Groups (Vol. 1)

Авторы: Gorenstein D., Lyons R., Solomon R.

Аннотация:

This book offers a single source of basic facts about the structure of the finite simple groups with emphasis on a detailed description of their local subgroup structures, coverings and automorphisms. The method is by examination of the specific groups, rather than by the development of an abstract theory of simple groups. While the purpose of the book is to provide the background for the proof of the classification of the finite simple groups — dictating the choice of topics — the subject matter is covered in such depth and detail that the book should be of interest to anyone seeking information about the structure of the finite simple groups. This volume offers a wealth of basic facts and computations. Much of the material is not readily available from any other source. In particular, the book contains the statements and proofs of the fundamental Borel-Tits Theorem and Curtis-Tits Theorem. It also contains complete information about the centralizers of semisimple involutions in groups of Lie type, as well as many other local subgroups.

Язык:

Рубрика: Математика/Алгебра/Теория групп/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 165

Добавлена в каталог: 26.03.2005

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Предметный указатель

-property      24 28 40—41 62 123 127
-property, partial      30 36 42 64
$G \aprox G*$       59 62—63 67—68 76—77 83
$G \aprox G*$ , , $n \geq 13$       121
$G \aprox G*$ , , $n \leq 12$       109 114
$G \aprox G*$ , G* of Lie type, of large Lie rank      116 121
$G \aprox G*$ , G* of Lie type, of small Lie rank, of characteristic 2      95—96 115
$G \aprox G*$ , G* of Lie type, of small Lie rank, of characteristic 2 and Lie rank 1      95—96
$G \aprox G*$ , G* of Lie type, of small Lie rank, of odd characteristic      110 112 113
$G \aprox G*$ , G* sporadic      109
$L_{p'}$ -balance      21 127—128
$L_{p'}$ -balance, analogue for near components      96
$L_{p'}$ -balance, analogue for two primes      134
$L_{p'}*$ -balance      127—128
$\mathcal{C}_p$ -groups      54 57 81 99—101
$\mathcal{C}_p$ -groups as pumpups      101—102
$\mathcal{C}_{p'}$ -groups      95 100
$\mathcal{G}_p$ -groups      57—58 63 103
$\mathcal{K}$ -groups      5 12
$\mathcal{K}$ -groups, theory of      12 45 48 75—76 138—139
$\mathcal{K}$ -proper group      12 75—76 79
$\mathcal{T}_p$ -groups      57 102—103 129
$\mathcal{T}_p$ -groups as pumpups      101—103
$\sigma(G)$       37—38 52—53 55 58—59 61 82 86 92—93 105 131
$\sigma_0(G)$       58—61 83 86 105—106 116 118 131
(B, N)-pair      34
(B, N)-pair, split      34
(B, N)-pair, split, recognition of rank 1      36—37 39 49 50 63 113 138
(B, N)-pair, split, recognition of rank 2      37 63 111 113 115 137—138
2-central involution      88
2-local p-rank      135
3/2-balanced functor      43 64—65
Algebraic automorphism      118 121
Alperin, J.      39 41
Amalgam method      5—6 26 39 41 43 60—61 105 131—133
Amalgam method, associated graph      132—133
Artin, E.      11
Aschbacher $\chi$ -block      39 41 53
Aschbacher, M.      18 30 37 39—43 45—48 50 53 89 99 125 129 130
Associated $(k + \frac12)$ -balanced functor      125
Atlas of Finite Groups      45 50 139
Background references      47—50 140
Background results      59 63 79 87 104 118
Background results, listed      44—50
Balance, k-balance      see “Group” (see also “Signalizer functor”)
Bar convention      18 139
Baumann, B.      39 131
Bender method      30 38 43 60 62 104 110 123 134
Bender, H.      16—17 48—50 123
Blackburn, N.      46 47
Bombieri, E.      49
Borel, A.      25
Brauer — Suzuki theory of exceptional characters      38 135
Brauer, R.      51
Brauer, R., theory of blocks      38 46 50 62
Brauer, R., theory of blocks, defect groups of 2-rank at most 3      50
Building      34 73 138
Burnside, W.      29—30
Carter, R.      45 47
Centralizer of element of odd prime order p      35 41 42 51 54—56
Centralizer of element of prime order p      108
Centralizer of involution      11 27ff. 35 39 41—43 51 52 54 61—62
Centralizer of involution pattern      46 59 77 109—110
Centralizer of semisimple element      51 54—56
Character theory      31 46 50 60 62 104 108 135—137
Character theory, ordinary vs.modular      50
Chevalley groups      see “Groups of Lie type”
Chevalley, C.      3
Chief factor, series      13—14
Classical groups      6ff. (see also “Groups of Lie type”)
Classification Grid      79 83 85 99—121
Classification Theorem      see “Theorems”
Classification Theorem, Theorems $\mathcal{C}_1$ - $\mathcal{C}_7$       104—106
Component      17 51 81
Component, solvable      51 67 109
Component, standard      53 91—92
Component, terminal      23 42 53 81 90—92 108
Composition factor, length, series      12
Composition factor, length, series, A-composition factor, length, series      13
Computer      35 45 68
Control of 2-locals      129
Control of fusion      87 122
Control of rank 1 or 2 fusion      91—92
Conway, J.      11
Core      20 (see also “p'-core”)
Core, elimination      40 43 60 110—111
Covering group      16
Covering group, notation for      101
Covering group, universal      17 33
Curtis, C.      35
Das, K.M.      35 71
Delgado, A.      37
Dickson, L.      7
Dieudonne, J.      47
Double transitivity of Suzuki type      95—96
Enguehard, M.      49—50
Expository references      47 141—146
Feit, W.      46 47 48 107—108
Finkelstein, L.      35
Fischer, B.      11 39—40
Fischer, B., transpositions      11 39
Fitting length, series      19
Fitting subgroup      16
Fitting subgroup, generalized Fitting subgroup      17 123
Foote, R.      5 38 53 98
Frattini subgroup      18
Frobenius group      107
Frobenius, G.      29
Frohardt, D.      35
Fusion      29 60 62 63 104 120 122
Fusion, extremal conjugation      122
General local group theory      45—48
Generalized Fitting subgroup      17 123
Geometry associated with a finite group      35 73—74
Gilman, R.      35 39 41
Glauberman, G.      21 38 39 48—50 124 130
Goldschmidt, D.      39 43 49 125
Gomi, K.      37
Gorenstein, D.      29 38 39 41 46 47—50 99 124 126 127
Griess, R.L.      17 35 41 45
Group of Lie type      see “Groups of Lie type”
Group order formulas      50 135—137
Group, almost simple      18
Group, alternating      see “Alternating group”
Group, covering      16
Group, covering, notation for      101
Group, covering, universal      17 33
Group, DC-proper      12 21
Group, k-balanced      124—125 129
Group, k-balanced, $(k + \frac12)$ -balanced      125 129
Group, k-balanced, locally balanced      128
Group, k-balanced, locally k-balanced, $(k + \frac12)$ -balanced      126
Group, k-balanced, weakly /c-balanced, weakly locally /c-balanced      124 126
Group, nilpotent      15—16
Group, p-constrained      20
Group, p-solvable      24
Group, perfect      16
Group, quasisimple      16 53 81
Group, semisimple      16
Group, simple, table of      8—10
Group, solvable      13 16 73—74
Group, sporadic      see “Sporadic group”
Groups in $\mathcal{C}_p$       see “ $\mathcal{C}_p$ -groups”
Groups in $\mathcal{C}_{p'}$       see “ $\mathcal{C}_{p'}$ -groups”
Groups in $\mathcal{G}_p$       see “ $\mathcal{G}_p$ -groups”
Groups in $\mathcal{T}_p$       see “ $\mathcal{T}_p$ -groups”
Groups of 3/2-balanced type      120—121
Groups of characteristic 2-type      55 98 116—117
Groups of characteristic p-type      25 116—117
Groups of even type      36 53—55 57—59 81 87 98

Groups of even type, groups of restricted even type      58—59 82 90 95 99 102 106
Groups of G*-type (G* a target group), groups of 2-amalgam G*-type, $G* \in \mathcal{K}^{(4)}$       115
Groups of G*-type (G* a target group), groups of 2-central G*-type, $G* \in \mathcal{K}^{(2)}$ of Lie type      111
Groups of G*-type (G* a target group), groups of 2-maximal G*-type mod cores, $G* \in \mathcal{K}^{(2)}*$       111
Groups of G*-type (G* a target group), groups of 2-terminal G*-type, $G* \in \mathcal{K}^{(2)}*$       110
Groups of G*-type (G* a target group), groups of doubly transitive G*-type, , ,       112
Groups of generic type      35—36 55—59 63—68 79 106 118—121
Groups of generic type, groups of $\mathcal{L}_p$ -generic type      57—59 61
Groups of generic type, groups of generic even type      106
Groups of generic type, groups of generic odd type      106
Groups of generic type, groups of semisimple type      121
Groups of generic type, groups of semisimple type, groups of proper semisimple type      121
Groups of GF(2)-type      40—43
Groups of large sporadic type      37—38 61
Groups of Lie type      6ff. 32ff. 45—49
Groups of Lie type, as (B, N)-pair      34
Groups of Lie type, Borel subgroup of      33
Groups of Lie type, Bruhat decomposition of      34
Groups of Lie type, Cartan subgroup of      33
Groups of Lie type, Chevalley commutator formula      32
Groups of Lie type, Chevalley group      3
Groups of Lie type, Dynkin diagram of      32 71
Groups of Lie type, generation of      48 65—66
Groups of Lie type, Lie rank of twisted      32
Groups of Lie type, Lie rank of untwisted      32
Groups of Lie type, monomial subgroup of      34
Groups of Lie type, monomial subgroup of, reduced      34
Groups of Lie type, parabolic subgroup of      26 62
Groups of Lie type, rank 1, subgroup of      33 35 71
Groups of Lie type, Ree group      3 10 49 50
Groups of Lie type, root subgroup of      32
Groups of Lie type, root system      32
Groups of Lie type, root system, fundamental system      32
Groups of Lie type, Schur multiplier of      45 48
Groups of Lie type, semisimple element of      51
Groups of Lie type, Steinberg presentation of, relations      33
Groups of Lie type, Steinberg variation      7
Groups of Lie type, Suzuki group      3 10
Groups of Lie type, universal version of      33
Groups of Lie type, universal version of, and universal covering group      33
Groups of Lie type, untwisted      see “Chevalley group”
Groups of Lie type, Weyl group of      33 35
Groups of odd order, groups of $\sigma*(G)$ -uniqueness type      107
Groups of odd order, groups of odd order uniqueness type      107
Groups of odd order, groups of {p, q}-parabolic type      107—108
Groups of odd type      58—59 81 86
Groups of quasithin type      82 105 114—116
Groups of quasithin type, groups of 2-amalgam type      114—115
Groups of quasithin type, thin subcase      37 41
Groups of special type      36—38 58—61 79 103—106 106—108 110—118
Groups of special type, groups of $\mathcal{L}_2$ -special type      58—60 103—104
Groups of special type, groups of $\mathcal{L}_2$ -special type, groups of $\mathcal{L}\mathcal{B}_2$ -type      61—63 83 104 110—113 123
Groups of special type, groups of $\mathcal{L}_2$ -special type, groups of $\mathcal{L}\mathcal{B}_2$ -type, -subcase      61—63
Groups of special type, groups of $\mathcal{L}_2$ -special type, groups of $\mathcal{L}\mathcal{T}_2$ -type      104 114
Groups of special type, groups of $\mathcal{L}_2$ -special type, groups of $\mathcal{L}\mathcal{T}_2$ -type, groups of 2-terminal $\mathcal{L}\mathcal{T}_2$ -type      114
Groups of special type, groups of $\mathcal{L}_p$ -special type      58—60 105—106
Groups of special type, groups of $\mathcal{L}_p$ -special type, groups of $\mathcal{L}\mathcal{C}_p$ -type      105 116—118
Groups of special type, groups of $\mathcal{L}_p$ -special type, groups of $\mathcal{L}\mathcal{T}_p$ -type      106 118
Groups of special type, groups of $\mathcal{L}_p$ -special type, groups of $\mathcal{L}\mathcal{T}_p$ -type, groups of p-terminal $\mathcal{L}\mathcal{T}_p$ -type      118
Groups of special type, groups of $\mathcal{L}_p$ -special type, groups of quasisymplectic $\mathcal{L}\mathcal{C}_p$ -type      117
Groups of special type, groups of $\mathcal{L}_p$ -special type, groups of wide $\mathcal{L}\mathcal{C}_p$ -type      116 133—135
Groups of special type, groups of $\sigma*(G)$ -uniqueness type      107
Groups of special type, groups of odd order uniqueness type      107
Groups of special type, groups of special even type      105 106 114—118
Groups of special type, groups of special odd type      103—104 106—108 110—114
Groups of special type, groups of {p, q}-parabolic type      107—108
Groups with specified 2-structure, groups of 2-rank at most 2      36 135
Groups with specified 2-structure, groups of 2-rank at most 3      50
Groups with specified 2-structure, groups with semidihedral or wreathed Sylow 2-subgroups      76 136—137
Groups with specified 2-structure, groups with semidihedral or wreathed Sylow 2-subgroups, Brauer group order formula for regular groups      137
Groups with specified 2-structure, groups with semidihedral or wreathed Sylow 2-subgroups, characteristic power      136—137
Hall $\sigma$ -subgroup      46
Hall, M.      11
Hall, P.      16 25
Harada, K.      11 39 89
Hayashi, M.      37
Holt, D.      89
Hunt, D.      49
Huppert, B.      46 47
Identification of simple groups      see “Recognition of simple groups”
Involution fusion pattern      46 60 62 109 110
Isaacs, I.M.      46 47
Isomorphism question      11
Janko, Z.      11 49
Klinger — Mason method      61 105 116—118
Klinger, K.      38 116—117
Layer      17 (see also “p-layer”)
Leech lattice      11 35
Linear group      18
Local subgroup      19 (see also “p-local subgroup”)
Local subgroup, general structure of      27—28
Lyons, R.      41 48—50 99
Mason, G.      37—38 41 116—117
Mathieu, E.      11
Maximal subgroup      60 62—63 123
McBride, P.      30 49 124
Meierfrankenfeld, U.      130
Modules, failure of factorization      26 130 133
Modules, quadratic      25—26 123 130
N-group      38—39 73
Near component      50 96—97 129—131 133
Near component, alternating      96—97 130
Near component, associated module of      96
Near component, linear      96—97 130 131
Near component, standard      98
Near component, type       97
Neighbor      42 64—65 71 119
Neighbor, semisimple      64—65
Neighborhood      55—58 59 63—68 118—120
Neighborhood, $Z_6 \times Z_2$ -      61 117
Neighborhood, base of      119
Neighborhood, example of      56—57 68—70
Neighborhood, level      66 121
Neighborhood, span of      66—67 121
Neighborhood, vertical      65 76—77 118—120
Niles, R.      39 50
Notation      139
Odlyzko, A.      49
Outer automorphisms      18
Overall strategy of proof      35—38 42—43
O’Nan, M.      18 45 49 138
p'-core      19 (see also “Core”)
p'-core, elimination      23—24 37 61 120
p'-core, embedding of p’-core of p-local subgroups      21 127—128
p-central p-element      101
p-component      20
p-component preuniqueness hypothesis      91—92
p-component preuniqueness subgroup      see “Uniqueness subgroups”
p-component uniqueness theorems      30—31 38 53 65 90—92 118
p-component, p-terminal      22ff. 63 108—109
p-component, p-terminal, pumping up to      23 108—109
p-component, solvable      64 109
p-layer      20
p-local subgroup      19
p-local subgroup, embedding of p'-core of      20—21 127—128
p-local subgroup, embedding of p-layer of      21—24 127
p-source      64
Parts of the series      4—5 59 77—78 80
Parts of the series, Part II      38 52—53
Parts of the series, Part III      36
Parts of the series, Part IV      37
Parts of the series, Part V      37 38
perfect central extension      see “Covering group”
Permutation group      18 35
Permutation group, doubly transitive      74 112—113 138
Permutation group, doubly transitive of Suzuki type      95—96
Permutation group, doubly transitive, split (B,N)-pair of rank 1      95—96 112—113 138
Permutation group, highly transitive      11 35
Permutation group, of rank 3      11
Permutation group, representation as      31

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