Griess R.L.Jr. — Twelve sporadic groups :: Электронная библиотека попечительского совета мехмата МГУ

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Griess R.L.Jr. — Twelve sporadic groups

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Название: Twelve sporadic groups

Автор: Griess R.L.Jr.

Аннотация:

The finite simple groups come in several infinite families (alternating groups and the groups of Lie type) plus 26 sporadic groups. The sporadic groups, discovered between 1861 and 1975, exist because of special combinatorial or arithmetic circumstances. A single theme does not capture them all. Nevertheless, certain themes dominate. The 20 sporadics involved in the Monster, the largest sporadic group, constitute the Happy Family. A leisurely and rigorous study of two of their three generations is the purpose of this book. The level is suitable for graduate students with little background in general finite group theory, established mathematicians and mathematical physicists.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 172

Добавлена в каталог: 22.12.2013

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Предметный указатель

"Extra automorphism" of A      97
$(-1)^{X}$       36 95
$(A\times B) > 2$       22
$2\cdot Alt_{8}$       121
$2\cdot M_{12}$       76ff
$2^{1+24}_{+}$ , $2^{1+2k}_{+}$       117
$2^{1+8}_{+}$       117
$2^{12} M_{12}$       111
$2^{4}:GL(4,2)$       55 57
$5^{1+2}$       116
$Alt_{6}$       18 119
$Alt_{8}$       57 118
$Alt_{9}$       117
$A_{2}$ -lattice      90
$a_{i}$ , $b_{i}$ , $c_{i}$       59
$A_{k}(x)$       105
$A_{n}$ -lattice      91
$B=U\oplus V\oplus W$       132
$b_{i}$ , $c_{i}$ , $a_{i}$       59
$C_{1}$ , Aut( $C_{1}$ )      31 32
$C_{O_{1}}$ , simclicity of      101
$C_{O_{2}}$       123
$C_{O_{2}}$ , simplicity of      108
$C_{O_{3}}$       123
$C_{O_{3}}$ , double transitivity of      105
$C_{O_{3}}$ , simplicity of      105
$C_{\infty}$       132
$Dih_{8}$       11
$D_{0}$ , $E_{0}$ , $F_{+}$ , $F_{-}$       71
$D_{4}(2)$       121
$D_{n}$ -lattice      91
$Ext^{n}$       13
$E_{0}$ , $F_{+}$ , $F_{-}$ , $D_{0}$       71
$E_{6}$ -lattice      91
$E_{7}$ -lattice      91
$E_{8}$       134
$E_{8}$ -lattice      91
$E_{8}$ -lattice theta function      92
$Fi_{22}$ , $Fi_{23}$ , $Fi_{24}$       134
$F_{+}$ , $F_{-}$ , $D_{0}$ , $E_{0}$       71
$F_{1}$ , $\mathbb{M}$       1 131 134
$F_{2}$       134
$F_{3}$       134
$F_{5}$       134
$G_{2}(3)$       134
$HS_{16}$ -lattice theta function      92
$HS_{n}$       92
$H^{n}(\cdot,\cdot)$ , n-th cohomology group      12
$J_{1}$       135
$J_{2}=HJ$       104 123
$J_{3}$       135
$J_{4}$       137
$K_{ij}...$       44 77
$K_{i}$       39 77
$K_{J}$       44 77
$L_{0}$       95
$L_{1}$ , $L_{2}$       60
$L_{i,1}$       61
$l_{i}$       39
$M_{10}$       18 66
$M_{11}$       66 75
$M_{11}$ , $M_{12}$ , $M_{22}$ , $M_{23}$ , $M_{24}$       54ff
$M_{12}$       62ff 85ff 119 123
$M_{12}$ , classes of elements of order 3      87
$M_{22}$       75 114ff
$M_{23}$       75
$M_{24}$       2 6 18 19 47 54ff
$N_{24}$       97
$N_{24}$ , conjugacy of involutions      110
$N_{24}$ , maximality of      100
$O^{+}(8,2)$       117
$O^{\epsilon}(2n,q)$       21
$O_{2}$       30
$p^{1+2n}_{\varepsilon}$       20
$Quat_{8}$       11 119
$r_{0}$ , $r_{1}$ , $r_{\omega}$ , $r_{\hat{\omega}}$       57
$r_{ijkl}$       57
$SQ_{n}$       91
$Sym_{4}$       30
$Sym_{5}$       29
$Sym_{8}$       57
$S^{\pm}$       71
$S_{z}(8)$       136
$U=S^{2}(E)$       132
$W_{E_{6}}$       118
$x_{i}$ , $x_{j}$ , z      113
$z=\varepsilon_{\mathcal{O}}$       114
$Z\times S$       28
$\alpha$       33 117
$\alpha_{i}$       95
$\alpha_{L}$ , $\alpha_{R}$       64 86
$\alpha_{S}$ , for $S \subseteq \Omega$       95
$\beta$       86 117
$\beta_{L}$ , $\beta_{R}$       64
$\Delta$       118
$\Delta(\alpha)$       124
$\delta_{L}$ , $\delta_{R}$       64
$\eta_{i}$       97
$\eta_{T}$       97
$\frac{1}{2}+\mathbf{Z}$ , the semiintegers      93
$\gamma$       117
$\Gamma(a)$       124
$\gamma^{'}_{L}$ , $\gamma^{'}_{R}$       64
$\gamma_{L}$ , $\gamma_{R}$       64
$\lambda$ , $\rho$       84
$\Lambda$ , the standard Leech lattice      95
$\Lambda^{2}_{2}$ , $\Lambda^{3}_{2}$ , $\Lambda^{4}_{2}$       95
$\lambda_{*}$ , $\rho_{*}$       85
$\lambda_{n}$       95
$\Lambda_{\pm}$       118
$\mathbb{F}^{6}_{4}$       78
$\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$       26
$\mathbb{M}$ , $F_{1}$ , symbols for the Monster      1 131
$\mathbf{P}^{1}(23)$ , the projective line over the field of 23 elements, $\mathbf{P}^{1}(F)$ , the projective line over the field F      60
$\mathbf{P}^{1}(7)$ , the projective line over the field of 7 elements      59
$\mathbf{P}^{2}(4)$ , the projective plane over the field of 4 elements      66
$\mathcal{A}=\{\alpha_{i}|i\in \Omega\}$ , standard basis for $\mathbb{R}^{\Omega}$       95
$\mathcal{D}$ , a common symbol for dodecad      37ff
$\mathcal{H}$       30
$\mathcal{L}$       39
$\mathcal{O}$ , a common symbol for an octad      36
$\mathcal{SL}(m,q)$       71
$\mathcal{SL}_{2}$ , $\mathcal{SL}_{3}$ , $\mathcal{SL}_{4}$       48 68
$\mathcal{SL}_{k}$       48 68
$\mathcal{TG}$       78
$\mathcal{TG}:Aut(\mathcal{TG})$       121
$\mathcal{T}$       54ff
$\mathcal{W}$       40
$\nu_{i}$       95
$\omega$ , a primitive cube root of unity in $\mathbf{F}_{4}$       30
$\omega$ , multiplication by      42
$\Omega^{+}(8,2)$       117
$\Omega^{\eplsilon}(n,q)$       18 19
$\overline{\omega}$ , the other primitive cube root of unity in $\mathbf{F}_{4}$       30
$\Phi$ -labeled points      82
$\pi$       42
$\pi_{0}$       41 42
$\Pi_{1}$       42 57
$\rho$ , $\lambda$       84
$\rho_{*}$ , $\lambda_{*}$       85
$\sigma(P)$       9
$\Sigma_{7}$       116
$\Sigma_{8}$       122
$\Sigma_{L}$ , $\Sigma_{R}$ , $\Sigma_{X}$       83
$\tau$       26 64
$\tau_{L}^{'}$ , $\tau_{R}^{'}$       64 86
$\tilde{\alpha}$       85
$\tilde{\beta}$       86
$\tilde{\gamma}$       85
$\tilde{\tau}^{'}$       85

$\varepsilon_{S}$ , for $S\subseteq \Omega$       95
$\varepsilon_{\mathcal{O} + \Omega}=z$       117
$\vartheta_{1}$ , $\vartheta_{2}$ , $\vartheta_{3}$ , $\vartheta_{4}$       117
$^{2}E_{6}(2)$       133
((abcd))      78
(T)      78
.223      121
.223, simplicity of      107
.333      122
.abc, the stabilizer in Aut( $\Lambda$ ) of a triangle of type abc      101
2-rank      75
27th sporadic      138
4-tuple labeling      77 82
6-tuple labeling map      39
A.B, A:B, $A \cdot B$       11
abc, triangle of type      101
Accents graves      35
Additive functor      12
AGL(2,3)      66
Alperin, Jon      15 19
Alternating groups      16
Anisotropic      25
Annihilator      90
Associated graph      124
Associated labeled set      78
Associated picture      37
Augmentation submodule      4
Aut( $Alt_{6}$ )      116
Aut( $\matcal{TG}$ )      84ff 117 119
Aut(C), Aut*(C), for a code C      27
Aut(F)      26
Aut(L), for a lattice L      89
B      132ff
Baby Monster      133 134
Balance condition, well-balanced condition      40 78
Balance, hexacode      34
Balanced set, parity, well balanced      40 76
Benson, David      3 137
Bilinear      24 25
Bilinear function      21
BINARY      25
Binary Golay code      36
Boolean sum      24
Broue, Michel      135
Burnside Basis Theorem      8
Burnside Normal p-Complement Theorem      8
C=C(z), centralizer of the involution z      117 131ff
Central extension      14
Chevalley — Steinberg presentation      22
Choi, Chang      54
Classes of elements of order 3 in $M_{12}$       87
Classes of elements of order 5 in $M_{24}$       122
Classification of even unimodular lattices      93
Classification of finite simple groups      137
Classification theorems      125
Cocode      26 80
Code equivalence, strict equivalence      27
Code equivalence, strict equivalence, thickened codes      27
Code parameters      25
Code, linear error correcting codes, error correcting codes      25ff
Cohomology of groups      12
Column distribution      81
Commuting criterion      57
Conjugacy classes 1A, 2A, 2B, 3A, 3C, 4A, 4B, 5A, 6A      135
Conjugacy classes of elements of order 3 in Aut( $\Lambda$ )      120
Conjugacy of involution in $C_{O_{0}}$       111
Conjugacy of involutions in $N_{24}$       110
Conway groups, $C_{O_{0}}$ , $C_{O_{1}}$ , $C_{O_{2}}$ , $C_{O_{3}}$       101
Coordinate permutations      26
Covering      14
Coxeter group      22
Curran, Peter      19
Curtis, Robert      3 54
Cycle shape $1^{6}3^{6}$       116 118
Cycle shapes      116
Darstellungsgruppe      15
Defective      24
Degenerate      24
Degree      5
Diag(n,F), $Diag_{1}(n,F)$       26 27
Diagonal group      26
Diagonal matrices      68
Distance      25
Dodecad      37ff
Dodecad pure subspace      70
Dodecad triangle      52
Dodecad-universe pure subspace      71
Dodecads, left and right, L, R      64 86
Dodecads: subspaces with none      75
Elements of order 3      116ff
Elliptic modular function, j      135
Equivalent codes, strict linear equivalence      27
Error correcting codes      25
Error correcting properties of tetracode      77
Essential extension      14
Even elements      6
Existence corollary      134
Extended monomial group      26
Extension      11
Extension kernel      11
Extension quotient      11
Extension theory      3
Extension, simultaneous componentwise downward      22
Extension, simultaneous componentwise upward      22
Extra automorphism      133
Extraspecial p-group      19
Feit, Walter      3 8
Field automorphism      68
Finite groups of Lie type      16
Finiteness      133
Fischer — Griess Monster      1 131ff
Fischer — Griess simple group      1 131ff
Fischer, Bernd      1 133
Formula for the number of isotropic subspaces      27
Fours group, 2A-pure      133
Foursome      81
Frame      102
Frame in $E_{8}$       94
Frames fixed by $N_{24}$       112
Frattini argument      5
Frattini subgroup      8
Frohardt, Daniel      136
FSX(3,4)      55 67 75 108
functor      12
G( $S^{\pm}$ )      72ff
G( $\mathcal{O}$ )      55 57ff
G( $\mathcal{T}$ )      55 58
G( $\Xi$ )      55 57
G(i)      59
G(X), $R(X)=O_{2}(G(X))$ , L(X)      55
Generations      1
Genus zero      135
GL(2,112)2      137
GL(2,3)      76
GL(2,F)      60
GL(3,2)      14 137
GL(4,2)      57
GL(n,F)      9
GL(n,q)      6
Golay cocode      38
Golay code, binary Golay code, $\mathcal{G}$       27 36ff
Golay code, uniqueness of      44 46
Gorenstein, Daniel      19 137 138
Gram matrix      88
Graph      124
Griess algebra      135
Griess, Robert      15
Half spin lattice, $HS_{n}$       92
Hall — Janko group      104 123
Hamming code      27

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