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

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

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

јннотаци€:

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 Ч dictating the choice of topics Ч 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: en

–убрика: ћатематика/јлгебра/“еори€ групп/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1994

 оличество страниц: 165

ƒобавлена в каталог: 26.03.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$B_p$-property      24 28 40Ч41 62 123 127
$B_p$-property, partial      30 36 42 64
$G \aprox G*$      59 62Ч63 67Ч68 76Ч77 83
$G \aprox G*$, $G* = A_n$, $n \geq 13$      121
$G \aprox G*$, $G* = A_n$, $n \leq 12$      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      95Ч96 115
$G \aprox G*$, G* of Lie type, of small Lie rank, of characteristic 2 and Lie rank 1      95Ч96
$G \aprox G*$, G* of Lie type, of small Lie rank, of odd characteristic      110 112 113
$G \aprox G*$, G* sporadic      109
$L_{p'}$-balance      21 127Ч128
$L_{p'}$-balance, analogue for near components      96
$L_{p'}$-balance, analogue for two primes      134
$L_{p'}*$-balance      127Ч128
$\mathcal{C}_p$-groups      54 57 81 99Ч101
$\mathcal{C}_p$-groups as pumpups      101Ч102
$\mathcal{C}_{p'}$-groups      95 100
$\mathcal{G}_p$-groups      57Ч58 63 103
$\mathcal{K}$-groups      5 12
$\mathcal{K}$-groups, theory of      12 45 48 75Ч76 138Ч139
$\mathcal{K}$-proper group      12 75Ч76 79
$\mathcal{T}_p$-groups      57 102Ч103 129
$\mathcal{T}_p$-groups as pumpups      101Ч103
$\sigma(G)$      37Ч38 52Ч53 55 58Ч59 61 82 86 92Ч93 105 131
$\sigma_0(G)$      58Ч61 83 86 105Ч106 116 118 131
(B, N)-pair      34
(B, N)-pair, split      34
(B, N)-pair, split, recognition of rank 1      36Ч37 39 49 50 63 113 138
(B, N)-pair, split, recognition of rank 2      37 63 111 113 115 137Ч138
2-central involution      88
2-local p-rank      135
3/2-balanced functor      43 64Ч65
Algebraic automorphism      118 121
Alperin, J.      39 41
Amalgam method      5Ч6 26 39 41 43 60Ч61 105 131Ч133
Amalgam method, associated graph      132Ч133
Artin, E.      11
Aschbacher $\chi$-block      39 41 53
Aschbacher, M.      18 30 37 39Ч43 45Ч48 50 53 89 99 125 129 130
Associated $(k + \frac12)$-balanced functor      125
Atlas of Finite Groups      45 50 139
Background references      47Ч50 140
Background results      59 63 79 87 104 118
Background results, listed      44Ч50
Balance, k-balance      see УGroupФ (see also УSignalizer functorФ)
Bar convention      18 139
Baumann, B.      39 131
Bender method      30 38 43 60 62 104 110 123 134
Bender, H.      16Ч17 48Ч50 123
Blackburn, N.      46 47
Bombieri, E.      49
Borel, A.      25
Brauer Ч 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.      29Ч30
Carter, R.      45 47
Centralizer of element of odd prime order p      35 41 42 51 54Ч56
Centralizer of element of prime order p      108
Centralizer of involution      11 27ff. 35 39 41Ч43 51 52 54 61Ч62
Centralizer of involution pattern      46 59 77 109Ч110
Centralizer of semisimple element      51 54Ч56
Character theory      31 46 50 60 62 104 108 135Ч137
Character theory, ordinary vs.modular      50
Chevalley groups      see УGroups of Lie typeФ
Chevalley, C.      3
Chief factor, series      13Ч14
Classical groups      6ff. (see also УGroups of Lie typeФ)
Classification Grid      79 83 85 99Ч121
Classification Theorem      see УTheoremsФ
Classification Theorem, Theorems $\mathcal{C}_1$-$\mathcal{C}_7$      104Ч106
Component      17 51 81
Component, solvable      51 67 109
Component, standard      53 91Ч92
Component, terminal      23 42 53 81 90Ч92 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      91Ч92
Conway, J.      11
Core      20 (see also Уp'-coreФ)
Core, elimination      40 43 60 110Ч111
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      95Ч96
Enguehard, M.      49Ч50
Expository references      47 141Ч146
Feit, W.      46 47 48 107Ч108
Finkelstein, L.      35
Fischer, B.      11 39Ч40
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      45Ч48
Generalized Fitting subgroup      17 123
Geometry associated with a finite group      35 73Ч74
Gilman, R.      35 39 41
Glauberman, G.      21 38 39 48Ч50 124 130
Goldschmidt, D.      39 43 49 125
Gomi, K.      37
Gorenstein, D.      29 38 39 41 46 47Ч50 99 124 126 127
Griess, R.L.      17 35 41 45
Group of Lie type      see УGroups of Lie typeФ
Group order formulas      50 135Ч137
Group, almost simple      18
Group, alternating      see УAlternating groupФ
Group, covering      16
Group, covering, notation for      101
Group, covering, universal      17 33
Group, DC-proper      12 21
Group, k-balanced      124Ч125 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      15Ч16
Group, p-constrained      20
Group, p-solvable      24
Group, perfect      16
Group, quasisimple      16 53 81
Group, semisimple      16
Group, simple, table of      8Ч10
Group, solvable      13 16 73Ч74
Group, sporadic      see УSporadic groupФ
Groups in $\mathcal{C}_p$      see У$\mathcal{C}_p$-groupsФ
Groups in $\mathcal{C}_{p'}$      see У$\mathcal{C}_{p'}$-groupsФ
Groups in $\mathcal{G}_p$      see У$\mathcal{G}_p$-groupsФ
Groups in $\mathcal{T}_p$      see У$\mathcal{T}_p$-groupsФ
Groups of 3/2-balanced type      120Ч121
Groups of characteristic 2-type      55 98 116Ч117
Groups of characteristic p-type      25 116Ч117
Groups of even type      36 53Ч55 57Ч59 81 87 98
Groups of even type, groups of restricted even type      58Ч59 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, $G* = L_3(p^n)$, $U_3(p^n)$, $^2G_2(3^n)$      112
Groups of generic type      35Ч36 55Ч59 63Ч68 79 106 118Ч121
Groups of generic type, groups of $\mathcal{L}_p$-generic type      57Ч59 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      40Ч43
Groups of large sporadic type      37Ч38 61
Groups of Lie type      6ff. 32ff. 45Ч49
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 65Ч66
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 УChevalley groupФ
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      107Ч108
Groups of odd type      58Ч59 81 86
Groups of quasithin type      82 105 114Ч116
Groups of quasithin type, groups of 2-amalgam type      114Ч115
Groups of quasithin type, thin subcase      37 41
Groups of special type      36Ч38 58Ч61 79 103Ч106 106Ч108 110Ч118
Groups of special type, groups of $\mathcal{L}_2$-special type      58Ч60 103Ч104
Groups of special type, groups of $\mathcal{L}_2$-special type, groups of $\mathcal{L}\mathcal{B}_2$-type      61Ч63 83 104 110Ч113 123
Groups of special type, groups of $\mathcal{L}_2$-special type, groups of $\mathcal{L}\mathcal{B}_2$-type, $SL_2 (q)$-subcase      61Ч63
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      58Ч60 105Ч106
Groups of special type, groups of $\mathcal{L}_p$-special type, groups of $\mathcal{L}\mathcal{C}_p$-type      105 116Ч118
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 133Ч135
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 114Ч118
Groups of special type, groups of special odd type      103Ч104 106Ч108 110Ч114
Groups of special type, groups of {p, q}-parabolic type      107Ч108
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 136Ч137
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      136Ч137
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 УRecognition of simple groupsФ
Involution fusion pattern      46 60 62 109 110
Isaacs, I.M.      46 47
Isomorphism question      11
Janko, Z.      11 49
Klinger Ч Mason method      61 105 116Ч118
Klinger, K.      38 116Ч117
Layer      17 (see also Уp-layerФ)
Leech lattice      11 35
Linear group      18
Local subgroup      19 (see also Уp-local subgroupФ)
Local subgroup, general structure of      27Ч28
Lyons, R.      41 48Ч50 99
Mason, G.      37Ч38 41 116Ч117
Mathieu, E.      11
Maximal subgroup      60 62Ч63 123
McBride, P.      30 49 124
Meierfrankenfeld, U.      130
Modules, failure of factorization      26 130 133
Modules, quadratic      25Ч26 123 130
N-group      38Ч39 73
Near component      50 96Ч97 129Ч131 133
Near component, alternating      96Ч97 130
Near component, associated module of      96
Near component, linear      96Ч97 130 131
Near component, standard      98
Near component, type $G_2(3^n)$      97
Neighbor      42 64Ч65 71 119
Neighbor, semisimple      64Ч65
Neighborhood      55Ч58 59 63Ч68 118Ч120
Neighborhood, $Z_6 \times Z_2$-      61 117
Neighborhood, base of      119
Neighborhood, example of      56Ч57 68Ч70
Neighborhood, level      66 121
Neighborhood, span of      66Ч67 121
Neighborhood, vertical      65 76Ч77 118Ч120
Niles, R.      39 50
Notation      139
Odlyzko, A.      49
Outer automorphisms      18
Overall strategy of proof      35Ч38 42Ч43
OТNan, M.      18 45 49 138
p'-core      19 (see also УCoreФ)
p'-core, elimination      23Ч24 37 61 120
p'-core, embedding of pТ-core of p-local subgroups      21 127Ч128
p-central p-element      101
p-component      20
p-component preuniqueness hypothesis      91Ч92
p-component preuniqueness subgroup      see УUniqueness subgroupsФ
p-component uniqueness theorems      30Ч31 38 53 65 90Ч92 118
p-component, p-terminal      22ff. 63 108Ч109
p-component, p-terminal, pumping up to      23 108Ч109
p-component, solvable      64 109
p-layer      20
p-local subgroup      19
p-local subgroup, embedding of p'-core of      20Ч21 127Ч128
p-local subgroup, embedding of p-layer of      21Ч24 127
p-source      64
Parts of the series      4Ч5 59 77Ч78 80
Parts of the series, Part II      38 52Ч53
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 УCovering groupФ
Permutation group      18 35
Permutation group, doubly transitive      74 112Ч113 138
Permutation group, doubly transitive of Suzuki type      95Ч96
Permutation group, doubly transitive, split (B,N)-pair of rank 1      95Ч96 112Ч113 138
Permutation group, highly transitive      11 35
Permutation group, of rank 3      11
Permutation group, representation as      31
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