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

Peterfalvi, T.      48—49 96 107 138
Phan, K.-W.      35 71
Presentation      31—35
Presentation of classical groups      35
Presentation of symmetric groups      32 36 68—70
Presentation, Steinberg presentation of groups of Lie type      32 33 36 51 118
Presentation, Steinberg presentation of groups of Lie type, a la Curtis — Tits      34 67 70
Presentation, Steinberg presentation of groups of Lie type, a la Gilman — Griess      35 67 71
Pumpup      22 127
Pumpup as $\mathcal{C}_p$ , $\mathcal{T}_p$ , or $\mathcal{G}_p$ -group      103 114
Pumpup, diagonal, proper, trivial, or vertical      22 127
Quasisimple group      16 53 81
Quasithin      5 37 41 43 60—61 82 105 114—116
Recognition of simple groups      31—35
Recognition of simple groups, alternating groups      32
Recognition of simple groups, groups of Lie type      32—35 42 49 137—138
Recognition of simple groups, recognition of symmetric groups      68—70
Recognition of simple groups, sporadic groups      35
Ree groups      3 10 49 50
Ree, R.      10
Rowley, P.      37
Schreier property      21 24 29
Schreier, O.      21
Schur multiplier      18 45 48 139
Schur, I.      17
Scott, L.      18
Section      12
Seitz, G.      48
Sibley, D.      48
Signalizer functor      29—30 124ff.
Signalizer functor method      29—30 36—38 41 60 64—65 104—105 116 118 120 128—129
Signalizer functor, closed      124
Signalizer functor, closure      30 124
Signalizer functor, complete      124
Signalizer functor, k-balanced, weakly k-balanced      124—126
Signalizer functor, solvable      124
Signalizer functor, trivial      124 128
Simple groups, table of      8—10
Simplicity criteria      29—31
Sims, C.      39 45
Sims, C., conjecture      74
Smith, F.      41
Smith, S.      41
Solomon, R.      35 39 53
Sporadic groups      3 6 9 11 87 100
Sporadic groups, as $\mathcal{G}_p$ -group      103
Sporadic groups, as target group      77 87
Sporadic groups, background properties of      44—48 50
Sporadic groups, existence and uniqueness of      46—47 50 72
Sporadic groups, individual groups, Baby Monster       11 38 87 100
Sporadic groups, individual groups, Conway groups $C_{0_1} = \dot 1$ , $C_{0_2} = \dot 2$ , $C_{0_3} = \dot 3$       11 38 42 72 87 100
Sporadic groups, individual groups, Fischer 3-transposition groups $F_{i_22}$ , $F_{i_23}$ , $F_{i'_24}$       11 38 87 100
Sporadic groups, individual groups, Fischer — Griess Monster       3 11 38 42 47 61 72 87 100
Sporadic groups, individual groups, Harada — Norton       11 67—68 87 100
Sporadic groups, individual groups, Held       11 87 100
Sporadic groups, individual groups, Higman — Sims HS      11 87 100
Sporadic groups, individual groups, Janko , , ,       3 11 87 100
Sporadic groups, individual groups, Lyons — Sims Ly      11 45 87 100
Sporadic groups, individual groups, Mathieu $M_{11}$ , $M_{12}$ , $M_{22}$ , $M_{23}$ , $M_{24}$       11 42 46 72 87 100
Sporadic groups, individual groups, McLaughlin Mc      11 87 100
Sporadic groups, individual groups, O’Nan-Sims O'N      11 87 100
Sporadic groups, individual groups, Rudvalis Ru      11 87 100
Sporadic groups, individual groups, Suzuki Suz      11 87 100
Sporadic groups, individual groups, Thompson       11 87 100
Standard form problems      38
Steinberg, R.      3 7 10 17 45 47 48
Stellmacher, B.      37 50
Strongly $\mathcal{Z}$ -embedded subgroup      see “Uniqueness subgroups”
Strongly closed subgroup      61
Strongly embedded subgroup      see “Uniqueness subgroups”
Strongly p-embedded subgroup      see “Uniqueness subgroups”
Stroth, G.      5 37 38 43 53 98—99 107 130
Subterminal pair      64—66
Suzuki groups       3 10
Suzuki, M.      10 38 46 47 49
Symmetric group $\Sigma_n$       6
Symplectic pair      117
Symplectic pair, faithful, of $\Omega_8^_(3)$ -type, trivial      117
Symplectic type 2-group      116
Tanaka, Y.      37
Target group (G*)      27 54—55 86—87
Target group (G*) in $K^{(0)$       89
Target group (G*) of Lie type      76—77
Target group (G*) of Lie type, $L_3^{\pm}(q)$ , , , , $L_4^{\pm}(q)$ , q odd      61—63
Target group (G*) of Lie type, of large Lie rank      63—68 71 76
Target group (G*) of Lie type, of Lie rank 1      77 110—113
Target group (G*) of Lie type, of small Lie rank      77 110—116
Target group (G*), , $n \geq 13$       63—68 76—77
Target group (G*), , $n \geq 13$ , $\Sigma_n$       68—70
Target group (G*), , $n \leq 12$       77 111—112 114—116
Target group (G*), quasithin      77
Target group (G*), sporadic      77 87
Target group (G*), sporadic, large sporadic      116—117
Target group (G*), the 8 families $\mathcal{K}^{(i)}$ ( $0 \leq i \geq 7$ )      86
Theorem $U(\sigma)$       5 86 92 106—108 129 131
Theorem $U(\sigma)$ , Corollary $U(\sigma)$       92
Theorem $U(\sigma)$ , stages      1—3 98—99
Theorem PS      92
Theorem SA      89 97
Theorem SE      89—90
Theorem SE, stages      1—2 95—96
Theorem SF      89
Theorem SZ      88—89
Theorem TS      92 114
Theorem U(2)      89 97 106
Theorems , ,       92
Theorems, $\mathcal{C}_1$ - $\mathcal{C}_7$       104—106
Theorems, Alperin — Brauer — Gorenstein — Walter classification of groups of 2-rank 2      39 41
Theorems, Alperin — Goldschmidt conjugation theorem      97 122

Theorems, Aschbacher classical involution theorem      41—43
Theorems, Aschbacher proper 2-generated core theorem      89
Theorems, Aschbacher uniqueness case theorem      43 92
Theorems, Aschbacher — Bender — Suzuki strongly Z-embedded subgroup theorem      88
Theorems, Aschbacher — Gilman component theorem      39 40 52 75
Theorems, Aschbacher — Gilman component theorem, generalized to odd primes      75—76
Theorems, Baumann — Glauberman — Niles theorem      50 130—131
Theorems, Baumann — Glauberman — Niles theorem, revision of      50
Theorems, Bender F*-theorem      17 138
Theorems, Bender uniqueness theorem      123
Theorems, Bender — Suzuki strongly embedded subgroup theorem      30—31 33 36 38 52
Theorems, Bender — Suzuki strongly embedded subgroup theorem for $\mathcal{K}$ -proper groups      75
Theorems, Bender — Thompson signalizer lemma      116
Theorems, Brauer — Suzuki quaternion theorem      52
Theorems, Burnside -theorem      30
Theorems, classification theorem      6 79
Theorems, classification theorem, comparison of new and old proofs      41—44 55 63 72 74—76 98
Theorems, classification theorem, four-part division      58—59
Theorems, classification theorem, main logic      106
Theorems, classification theorem, stages of the proof      61 106—121
Theorems, Curtis — Tits theorem      35 63 67 113 115
Theorems, Curtis — Tits theorem, variations for classical groups      35 63 113 115
Theorems, Feit — Thompson theorem      see “Odd Order Theorem”
Theorems, Fong — Seitz classification of split (B, N)-pairs of rank 2      63 138
Theorems, Gilman — Griess recognition theorem for groups of Lie type      35 67 71
Theorems, Glauberman Z*-theorem      30 31 36 38 43 48 79 122 135
Theorems, Glauberman ZJ-theorem      26 38
Theorems, global C(G, ?)-theorem      38 53
Theorems, Goldschmidt strongly closed abelian 2-subgroup theorem      43 89
Theorems, Gorenstein — Harada sectional 2-rank at most 4 theorem      39 42 43
Theorems, Gorenstein — Walter dihedral Sylow 2-subgroup classification      50 74
Theorems, Gorenstein — Walter dihedral Sylow 2-subgroup classification, revision by Bender and Glauberman      50 74
Theorems, Hall — Higman theorem B      26
Theorems, Holt’s Theorem      89
Theorems, Jordan — Holder theorem      12ff
Theorems, local C(X, T)-theorem      50 96—97 129—131
Theorems, Mason quasithin theorem      37 41
Theorems, odd order theorem      30 31 36 38—39 48 79 104 134 135
Theorems, odd order theorem, revision of      48 74 123
Theorems, p-component uniqueness theorems      30—31 38 53 65 90—92 118
Theorems, Schur — Zassenhaus theorem      24
Theorems, signalizer functor theorems      49 124
Theorems, Solomon maximal 2-component theorem      39
Theorems, theorem $\mathcal{C}_1$       104
Theorems, Theorem $\mathcal{C}_1$ , stages 1—4      106—108
Theorems, theorem $\mathcal{C}_2$       104
Theorems, Theorem $\mathcal{C}_2$ , stages 1—4      110—113
Theorems, theorem $\mathcal{C}_3$       104
Theorems, Theorem $\mathcal{C}_3$ , stages 1—3      113
Theorems, theorem $\mathcal{C}_4$       105
Theorems, Theorem $\mathcal{C}_4$ , stages 1—4      114—116
Theorems, theorem $\mathcal{C}_5$       105
Theorems, Theorem $\mathcal{C}_5$ , stages 1—4      116—118
Theorems, theorem $\mathcal{C}_6$       106
Theorems, Theorem $\mathcal{C}_6$ , stages 1—2      118
Theorems, theorem $\mathcal{C}_7$       106
Theorems, Theorem $\mathcal{C}_7$ , stages 1—5      118—121
Theorems, Theorem M(S)      5 90 99 129
Theorems, Theorem M(S), stages 1—3      97—98
Theorems, Thompson $A \times B$ -lemma      21
Theorems, Thompson dihedral lemma      134
Theorems, Thompson factorization theorem      26 130
Theorems, Thompson N-group classification theorem      38—40 43
Theorems, Thompson order formula      114 136
Theorems, Thompson replacement theorem      26 130
Theorems, Thompson transfer lemma      55 104 122 128
Theorems, Three subgroup lemma      17
Theorems, Timmesfeld root involution theorem      42—43
Theorems, Tits classification of spherical buildings of rank at least 3      73
Theorems, Walter classification of groups with abelian Sylow 2-subgroups      74
Theorems, Walter classification of groups with abelian Sylow 2-subgroups, revision by Bender      74
Thompson subgroup      26 130
Thompson, J.G.      37 39—40 43 48 49 107—108 130
Tightly embedded subgroup      39 43
Timmesfeld, F.G.      37 39 40
Tits system      see “(B N)-pair”
Tits, J.      10 25 34 137
TRANSFER      29 31 74 107
Triality      10
Ulmost strongly p-embedded subgroup      see “Uniqueness subgroups”
Ulternating group       6 32 36
Ulternating group as $\mathcal{C}_2$ , $\mathcal{T}_2$ , or $\mathcal{G}_2$ -group      103
Uniqueness case      5 38 43 53
Uniqueness grid      79 83 84 87—89
Uniqueness subgroups, $\mathcal{L}\mathcal{C}_p$ -uniqueness subgroup      94
Uniqueness subgroups, 2-uniqueness subgroup      82 87—89
Uniqueness subgroups, near component 2-local uniqueness subgroup      97
Uniqueness subgroups, p-component preuniqueness subgroup      90—93
Uniqueness subgroups, p-component preuniqueness subgroup of strongly closed type      93
Uniqueness subgroups, p-component preuniqueness subgroup, controlling rank 1 or 2 fusion      91—92
Uniqueness subgroups, p-component preuniqueness subgroup, standard      91—92
Uniqueness subgroups, p-component preuniqueness subgroup, wreathed      93
Uniqueness subgroups, p-uniqueness subgroup, p odd      52—53 82
Uniqueness subgroups, p-uniqueness subgroup, p odd, strong p-uniqueness subgroup      52—53 58 63 68 82—83 92—96 99 106
Uniqueness subgroups, strongly embedded subgroup      30—31 75 82 87—89 95—96
Uniqueness subgroups, strongly p-embedded subgroup      31 82 91—92
Uniqueness subgroups, strongly p-embedded subgroupm almost strongly p-embedded subgroup      91—94
Uniqueness subgroups, strongly p-embedded subgroupm almost strongly p-embedded subgroup, almost p-constrained      93
Uniqueness subgroups, strongly p-embedded subgroupm almost strongly p-embedded subgroup, of strongly closed type      93
Uniqueness subgroups, strongly p-embedded subgroupm almost strongly p-embedded subgroup, wreathed      93
Uniqueness subgroups, strongly Z-embedded subgroup      87—89 96
Uniqueness subgroups, weakly Z-embedded subgroup      89
Uniqueness subgroups, {2, p}-uniqueness subgroup      53 98—99
Walter, J.H.      29 38—39 124 126 127
Ward, H.      49
Weak closure method      43
Weakly Z-embedded subgroup      see “Uniqueness subgroups”
Wong, W.      35
Yoshida, T.      29

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