|
|
|
[推荐]王小云的密码人生 1/2视屏
能啊 再试下 |
|
[原创]群论的一些基础知识
G := Sym({ "a", "b", "c" }); G; g:=AutomorphismGroup(G); g; PermutationGroup(g); K0:=G!("a","b","c"); K1:=G!("c","b","a"); K2:=G!("c","a","b"); K3:=G!("c","b"); K4:=G!("c","a"); K5:=G!("c","b"); K6:=G!("b","a"); hom< G -> G | K2, K4 >; hom< G -> G | K2, K6 >; hom< G -> G | K0->K0, K0->K0 >; Symmetric group G acting on a set of cardinality 3 Order = 6 = 2 * 3 A group of automorphisms of GrpPerm: G, Degree 3, Order 2 * 3 Generators: Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 which maps: (c, b, a) |--> (c, a, b) (c, b) |--> (c, a) Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 which maps: (c, b, a) |--> (c, b, a) (c, b) |--> (b, a) Permutation group acting on a set of cardinality 3 Order = 6 = 2 * 3 (2, 3) (1, 2, 3) Endomorphism of GrpPerm: G, Degree 3, Order 2 * 3 induced by (c, b, a) |--> (c, a, b) (c, b) |--> (c, a) Endomorphism of GrpPerm: G, Degree 3, Order 2 * 3 induced by (c, b, a) |--> (c, a, b) (c, b) |--> (b, a) Endomorphism of GrpPerm: G, Degree 3, Order 2 * 3 induced by (c, a, b) |--> (c, a, b) (c, a, b) |--> (c, a, b) ConjugacyClasses(G); ClassMap(G); Exponent(G) ; NumberOfClasses(G) ; PowerMap(G) ; SubgroupLattice(G); Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 3 Rep (c, b) [3] Order 3 Length 2 Rep (c, b, a) Mapping from: GrpPerm: G to { 1 .. 3 } 6 3 Mapping from: Cartesian Product<{ 1 .. 3 }, Integer Ring> to { 1 .. 3 } Partially ordered set of subgroup classes ----------------------------------------- [4] Order 6 Length 1 Maximal Subgroups: 2 3 --- [3] Order 3 Length 1 Maximal Subgroups: 1 [2] Order 2 Length 3 Maximal Subgroups: 1 --- [1] Order 1 Length 1 Maximal Subgroups: SimpleSubgroups(G); SubgroupClasses(G ); Conjugacy classes of subgroups ------------------------------ [1] Order 1 Length 1 Permutation group acting on a set of cardinality 3 Order = 1 [2] Order 2 Length 3 Permutation group acting on a set of cardinality 3 Order = 2 (b, a) [3] Order 3 Length 1 Permutation group acting on a set of cardinality 3 Order = 3 (c, b, a) [4] Order 6 Length 1 Symmetric group G acting on a set of cardinality 3 Order = 6 = 2 * 3 (c, b, a) (c, b) G := Sym({ "a", "b", "c" }); G; A:=AutomorphismGroup(G); A; Holomorph(G) ; PermutationGroup(A); K0:=G!("a","b","c"); K1:=G!("c","b","a"); K2:=G!("c","a","b"); K3:=G!("c","b"); K4:=G!("c","a"); K5:=G!("c","b"); K6:=G!("b","a"); k24:=hom< G -> G | K2, K4 >; k24; k26:=hom< G -> G | K2, K6 >; k26; k00:=hom< G -> G | K0->K0, K0->K0 >; k00; Kernel(k24); Kernel(k26); sub<G | >; n:=NormalSubgroups(G); n; innAutomorphismGroup:=sub<G | ("c", "b", "a")>; innAutomorphismGroup; Order(innAutomorphismGroup); IsSubnormal(G, innAutomorphismGroup); quo<G |>; G / innAutomorphismGroup; Centre(G); Symmetric group G acting on a set of cardinality 3 Order = 6 = 2 * 3 A group of automorphisms of GrpPerm: G, Degree 3, Order 2 * 3 Generators: Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 which maps: (c, b, a) |--> (c, a, b) (c, b) |--> (c, a) Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 which maps: (c, b, a) |--> (c, b, a) (c, b) |--> (b, a) Permutation group acting on a set of cardinality 6 (1, 2, 3)(4, 5, 6) (1, 5)(2, 4)(3, 6) (2, 3)(5, 6) (4, 6, 5) Homomorphism of GrpPerm: G, Degree 3, Order 2 * 3 into GrpPerm: $, Degree 6 induced by (c, b, a) |--> (1, 2, 3)(4, 5, 6) (c, b) |--> (1, 5)(2, 4)(3, 6) Homomorphism of GrpPerm: $, Degree 6 into A group of automorphisms of GrpPerm: G, Degree 3, Order 2 * 3 induced by (1, 2, 3)(4, 5, 6) |--> Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 of order 1 (1, 5)(2, 4)(3, 6) |--> Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 of order 1 (2, 3)(5, 6) |--> Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 (4, 6, 5) |--> Automorphism of GrpPerm: G, Degree 3, Order 2 * 3 Permutation group acting on a set of cardinality 3 Order = 6 = 2 * 3 (2, 3) (1, 2, 3) Endomorphism of GrpPerm: G, Degree 3, Order 2 * 3 induced by (c, b, a) |--> (c, a, b) (c, b) |--> (c, a) Endomorphism of GrpPerm: G, Degree 3, Order 2 * 3 induced by (c, b, a) |--> (c, a, b) (c, b) |--> (b, a) Endomorphism of GrpPerm: G, Degree 3, Order 2 * 3 induced by (c, a, b) |--> (c, a, b) (c, a, b) |--> (c, a, b) Permutation group acting on a set of cardinality 3 Order = 1 Permutation group acting on a set of cardinality 3 Order = 1 Permutation group acting on a set of cardinality 3 Mapping from: GrpPerm: $, Degree 3 to GrpPerm: G Conjugacy classes of subgroups ------------------------------ [1] Order 1 Length 1 Permutation group acting on a set of cardinality 3 Order = 1 [2] Order 3 Length 1 Permutation group acting on a set of cardinality 3 Order = 3 (c, b, a) [3] Order 6 Length 1 Permutation group acting on a set of cardinality 3 Order = 6 = 2 * 3 (b, a) (c, b, a) Permutation group innAutomorphismGroup acting on a set of cardinality 3 (c, b, a) 3 true Permutation group acting on a set of cardinality 3 Order = 6 = 2 * 3 (1, 2, 3) (1, 2) Mapping from: GrpPerm: G to GrpPerm: $, Degree 3, Order 2 * 3 Composition of Mapping from: GrpPerm: G to GrpPerm: $, Degree 6, Order 2 * 3 and Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $, Degree 3, Order 2 * 3 Permutation group acting on a set of cardinality 2 Id($) (1, 2) Permutation group acting on a set of cardinality 3 Order = 1 |
|
[原创]群论的一些基础知识
K := QuadraticField(-5); K; K1 := QuadraticField(-51); K1; K := QuadraticField(-26); K; K := QuadraticField(-1136); K; K := QuadraticField(-50); K; C<i> := ComplexField(5); Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 51 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 26 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 71 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field 12.056 - 4.7278*i 71 4.00000000000000000000000000000 |
|
[原创]群论的一些基础知识
L:=Lattice("A", 4); L; LL:=Lattice("Lambda", 3); LL; sub<L | > ; sub<LL | > ; Lattice of rank 4 and degree 5 Determinant: 5 Factored Determinant: 5 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 0 -1 1 0) ( 0 0 0 -1 1) Lattice of rank 3 and degree 3 Determinant: 4 Factored Determinant: 2^2 Basis: (-1 -1 0) (-1 1 0) ( 0 -1 1) Lattice of rank 0 and degree 5 Determinant: 1 Mapping from: Lattice of rank 0 and degree 5 to Lat: L Lattice of rank 0 and degree 3 Determinant: 1 Mapping from: Lattice of rank 0 and degree 3 to Lat: LL http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ http://akpublic.research.att.com/~njas/lattices/index.html http://akpublic.research.att.com/~njas/lattices/index.html#An |
|
[原创]群论的一些基础知识
格这种结构都能构成群吗?格新名词比群还多:,群格,洞,深洞,格归约基 L:=Lattice("A", 4); L; LL:=Lattice("Lambda", 3); LL; BasisMatrix(L) ; BasisMatrix(LL) ; SS:=BaseRing(L); S:=BaseRing(LL); S; CoefficientRing(L); CoefficientRing(LL); CoordinateRing(L); CoordinateRing(LL); AmbientSpace(L); CoordinateSpace(L) ; Category(L) ; Type(L) ; AmbientSpace(L); CoordinateSpace(LL) ; Category(LL) ; Type(LL) ; Dimension(L); Dimension(LL); Rank(L); Rank(LL) ; Degree(L); Degree(LL); Content(L); Content(LL); Level(L) ; Level(LL) ; Determinant(L) ; Determinant(L) ; GramMatrix(L); GramMatrix(L); M:=InnerProductMatrix(L) ; M; InnerProductMatrix(LL) ; GramMatrix(M) ; Basis(L); Basis(LL); ChangeRing(L, S); ChangeRing(LL, SS); BaseChange(L, S); BaseChange(LL, SS); BaseExtend(L, S); BaseExtend(LL, SS); v3:=L . 3; v3; v33:=LL.5; v33; ElementToSequence(v3) ; ElementToSequence(v33) ; Minimum(L) ; Minimum(LL) ; Holes(L) ; Holes(L) ; DeepHoles(L); DeepHoles(LL); Genus(L) ; Genus(LL) ; SetVerbose("LLL", 3) ; LLL(L) ; LLL(LL) ; AutomorphismGroup(L); Lattice of rank 4 and degree 5 Determinant: 5 Factored Determinant: 5 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 0 -1 1 0) ( 0 0 0 -1 1) Lattice of rank 3 and degree 3 Determinant: 4 Factored Determinant: 2^2 Basis: (-1 -1 0) (-1 1 0) ( 0 -1 1) [-1 1 0 0 0] [ 0 -1 1 0 0] [ 0 0 -1 1 0] [ 0 0 0 -1 1] [-1 -1 0] [-1 1 0] [ 0 -1 1] Integer Ring Integer Ring Integer Ring Integer Ring Integer Ring Full Vector space of degree 5 over Rational Field Mapping from: Lat: L to Full Vector space of degree 5 over Rational Field given by a rule [no inverse] Full Vector space of degree 4 over Rational Field Inner Product Matrix: [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] Mapping from: Lat: L to Full Vector space of degree 4 over Rational Field given by a rule [no inverse] Lat Lat Full Vector space of degree 5 over Rational Field Mapping from: Lat: L to Full Vector space of degree 5 over Rational Field given by a rule [no inverse] Full Vector space of degree 3 over Rational Field Inner Product Matrix: [ 2 0 1] [ 0 2 -1] [ 1 -1 2] Mapping from: Lat: LL to Full Vector space of degree 3 over Rational Field given by a rule [no inverse] Lat Lat 4 3 4 3 5 3 1 1 5 8 5 5 [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] [1 0 0] [0 1 0] [0 0 1] [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] [ (-1 1 0 0 0), ( 0 -1 1 0 0), ( 0 0 -1 1 0), ( 0 0 0 -1 1) ] [ (-1 -1 0), (-1 1 0), ( 0 -1 1) ] Lattice of rank 4 and degree 5 Determinant: 5 Factored Determinant: 5 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 0 -1 1 0) ( 0 0 0 -1 1) Lattice of rank 3 and degree 3 Determinant: 4 Factored Determinant: 2^2 Basis: (-1 -1 0) (-1 1 0) ( 0 -1 1) Lattice of rank 4 and degree 5 Determinant: 5 Factored Determinant: 5 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 0 -1 1 0) ( 0 0 0 -1 1) Mapping from: Lat: L to Lat: L Lattice of rank 3 and degree 3 Determinant: 4 Factored Determinant: 2^2 Basis: (-1 -1 0) (-1 1 0) ( 0 -1 1) Mapping from: Lat: LL to Lat: LL Lattice of rank 4 and degree 5 Determinant: 5 Factored Determinant: 5 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 0 -1 1 0) ( 0 0 0 -1 1) Mapping from: Lat: L to Lat: L Lattice of rank 3 and degree 3 Determinant: 4 Factored Determinant: 2^2 Basis: (-1 -1 0) (-1 1 0) ( 0 -1 1) Mapping from: Lat: LL to Lat: LL ( 0 0 -1 1 0) >> v33:=LL.5; ^ Runtime error in '.': Argument 2 (5) should be in the range [1 .. 3] >> v33; ^ User error: Identifier 'v33' has not been declared or assigned [ 0, 0, -1, 1, 0 ] >> ElementToSequence(v33) ; ^ User error: Identifier 'v33' has not been declared or assigned 2 2 [ ( 2/5 -3/5 2/5 2/5 -3/5), ( 3/5 -2/5 3/5 -2/5 -2/5), ( 3/5 -2/5 -2/5 3/5 -2/5), ( 1/5 -4/5 1/5 1/5 1/5), (-4/5 1/5 1/5 1/5 1/5), (-3/5 2/5 -3/5 2/5 2/5), ( 2/5 -3/5 -3/5 2/5 2/5), ( 3/5 3/5 -2/5 -2/5 -2/5), (-2/5 3/5 -2/5 -2/5 3/5), (-3/5 2/5 2/5 -3/5 2/5), (-2/5 3/5 -2/5 3/5 -2/5), (-3/5 -3/5 2/5 2/5 2/5), (-2/5 3/5 3/5 -2/5 -2/5), (-3/5 2/5 2/5 2/5 -3/5), ( 3/5 -2/5 -2/5 -2/5 3/5), (-1/5 4/5 -1/5 -1/5 -1/5), (-2/5 -2/5 3/5 -2/5 3/5), ( 2/5 2/5 2/5 -3/5 -3/5), ( 2/5 -3/5 2/5 -3/5 2/5), (-2/5 -2/5 -2/5 3/5 3/5), ( 2/5 2/5 -3/5 2/5 -3/5), (-1/5 -1/5 4/5 -1/5 -1/5), (-2/5 -2/5 3/5 3/5 -2/5), ( 1/5 1/5 -4/5 1/5 1/5), ( 2/5 2/5 -3/5 -3/5 2/5), ( 1/5 1/5 1/5 1/5 -4/5), (-1/5 -1/5 -1/5 4/5 -1/5), ( 1/5 1/5 1/5 -4/5 1/5), ( 4/5 -1/5 -1/5 -1/5 -1/5), (-1/5 -1/5 -1/5 -1/5 4/5) ] [ ( 2/5 -3/5 2/5 2/5 -3/5), ( 3/5 -2/5 3/5 -2/5 -2/5), ( 3/5 -2/5 -2/5 3/5 -2/5), ( 1/5 -4/5 1/5 1/5 1/5), (-4/5 1/5 1/5 1/5 1/5), (-3/5 2/5 -3/5 2/5 2/5), ( 2/5 -3/5 -3/5 2/5 2/5), ( 3/5 3/5 -2/5 -2/5 -2/5), (-2/5 3/5 -2/5 -2/5 3/5), (-3/5 2/5 2/5 -3/5 2/5), (-2/5 3/5 -2/5 3/5 -2/5), (-3/5 -3/5 2/5 2/5 2/5), (-2/5 3/5 3/5 -2/5 -2/5), (-3/5 2/5 2/5 2/5 -3/5), ( 3/5 -2/5 -2/5 -2/5 3/5), (-1/5 4/5 -1/5 -1/5 -1/5), (-2/5 -2/5 3/5 -2/5 3/5), ( 2/5 2/5 2/5 -3/5 -3/5), ( 2/5 -3/5 2/5 -3/5 2/5), (-2/5 -2/5 -2/5 3/5 3/5), ( 2/5 2/5 -3/5 2/5 -3/5), (-1/5 -1/5 4/5 -1/5 -1/5), (-2/5 -2/5 3/5 3/5 -2/5), ( 1/5 1/5 -4/5 1/5 1/5), ( 2/5 2/5 -3/5 -3/5 2/5), ( 1/5 1/5 1/5 1/5 -4/5), (-1/5 -1/5 -1/5 4/5 -1/5), ( 1/5 1/5 1/5 -4/5 1/5), ( 4/5 -1/5 -1/5 -1/5 -1/5), (-1/5 -1/5 -1/5 -1/5 4/5) ] [ ( 2/5 -3/5 2/5 2/5 -3/5), ( 3/5 -2/5 3/5 -2/5 -2/5), ( 3/5 -2/5 -2/5 3/5 -2/5), (-3/5 2/5 -3/5 2/5 2/5), ( 2/5 -3/5 -3/5 2/5 2/5), ( 3/5 3/5 -2/5 -2/5 -2/5), (-2/5 3/5 -2/5 -2/5 3/5), (-3/5 2/5 2/5 -3/5 2/5), (-2/5 3/5 -2/5 3/5 -2/5), (-3/5 -3/5 2/5 2/5 2/5), (-2/5 3/5 3/5 -2/5 -2/5), (-3/5 2/5 2/5 2/5 -3/5), ( 3/5 -2/5 -2/5 -2/5 3/5), (-2/5 -2/5 3/5 -2/5 3/5), ( 2/5 2/5 2/5 -3/5 -3/5), ( 2/5 -3/5 2/5 -3/5 2/5), (-2/5 -2/5 -2/5 3/5 3/5), ( 2/5 2/5 -3/5 2/5 -3/5), (-2/5 -2/5 3/5 3/5 -2/5), ( 2/5 2/5 -3/5 -3/5 2/5) ] [ ( 0 0 -1), (0 1 0), ( 0 -1 0), (-1 0 0), (1 0 0), (0 0 1) ] Genus of Lattice of rank 4 and degree 5 Determinant: 5 Factored Determinant: 5 Minimum: 2 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 0 -1 1 0) ( 0 0 0 -1 1) Genus of Lattice of rank 3 and degree 3 Determinant: 4 Factored Determinant: 2^2 Minimum: 2 Basis: (-1 -1 0) (-1 1 0) ( 0 -1 1) Lattice of rank 4 and degree 5Integer Gram Matrix construction; rows: 4, columns: 5 Time: 0.000 Basis-matrix based variant of L^3 on a (4 x 5) matrix. -> L^3-parameters: (9.995000E-01, 5.005000E-01) -> Working precision: 53 -> Using C doubles within orthogonalization. -> Using genuine Lovasz condition. 2/4. Step: 1 [0/0], Lovasz tests: 0, 0.000, Max Norm: 1.41421 3/4. Step: 2 [1/1], Lovasz tests: 1, 0.000, Max Norm: 1.41421 4/4. Step: 3 [1/1], Lovasz tests: 2, 0.000, Max Norm: 1.41421 Number of loop iterations: 3. Time: 0.000, Max Basis entry: 1.41421E+00 Total time for LLL: 0.000 Integer Gram Matrix construction; rows: 4, columns: 5 Time: 0.000 Determinant: 5 Factored Determinant: 5 Basis: (-1 1 0 0 0) ( 0 -1 1 0 0) ( 0 -1 0 1 0) ( 0 -1 0 0 1) [1 0 0 0] [0 1 0 0] [0 1 1 0] [0 1 1 1] Lattice of rank 3 and degree 3Integer Gram Matrix construction; rows: 3, columns: 3 Time: 0.000 Basis-matrix based variant of L^3 on a (3 x 3) matrix. -> L^3-parameters: (9.995000E-01, 5.005000E-01) -> Working precision: 53 -> Using C doubles within orthogonalization. -> Using genuine Lovasz condition. 2/3. Step: 1 [0/0], Lovasz tests: 0, 0.000, Max Norm: 1.41421 3/3. Step: 2 [1/1], Lovasz tests: 1, 0.000, Max Norm: 1.41421 Number of loop iterations: 2. Time: 0.000, Max Basis entry: 1.41421E+00 Total time for LLL: 0.000 Integer Gram Matrix construction; rows: 3, columns: 3 Time: 0.000 Determinant: 4 Factored Determinant: 2^2 Basis: (-1 -1 0) ( 0 -1 1) (-1 0 1) [1 0 0] [0 0 1] [0 1 1] Gram-matrix based variant of L^3 on a (4 x 4) matrix. -> L^3-parameters: (9.995000E-01, 5.005000E-01) -> Working precision: 53 -> Using C doubles within orthogonalization. -> Using genuine Lovasz condition. 2/4. Step: 1 [0/0], Lovasz tests: 0, 0.000, Max Norm: 1.41421 3/4. Step: 2 [1/1], Lovasz tests: 1, 0.000, Max Norm: 1.41421 4/4. Step: 3 [1/1], Lovasz tests: 2, 0.000, Max Norm: 1.41421 Number of loop iterations: 3. Time: 0.000, Max Norm: 1.41421 Total time for LLL: 0.000 MatrixGroup(4, Integer Ring) of order 2^4 * 3 * 5 Generators: [ 0 0 -1 -1] [ 0 -1 0 0] [ 0 1 1 0] [-1 -1 -1 0] [-1 0 0 0] [ 1 1 1 0] [ 0 0 0 1] [ 0 0 -1 -1] [ 1 0 0 0] [-1 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] |
|
[原创]群论的一些基础知识
GOOGLE.COM:不是Jacobson是Jacobi ,我以为能学20世纪的数学书了,一搜Jacobi19世纪的,Jacobson的书在这下了本,天书!http://ishare.iask.sina.com.cn/f/14751553.html?retcode=0 还有本格论的------汉语的少啊 收藏! Jacobi an 雅可比椭圆函数 雅可比矩阵 雅可比符号 雅可比恒等式 雅可比旋转 1804年10月4日-1851年2月18日)是一位普鲁士数学家,被广泛的认为是历史上最伟大的数学家之一。 Nathan Jacobson http://www-history.mcs.st-and.ac.uk/history/Biographies/Jacobson.html Born: 8 Sept 1910 in Warsaw, Russian Empire (now Poland) Died: 5 Dec 1999 in Hamden, Connecticut, USA Jacobson Theory of Rings Jacobson Structure of Rings Obituary: The New York Times Honours awarded to Nathan Jacobson (Click below for those honoured in this way) AMS Colloquium Lecturer 1955 LMS Honorary Member 1972 American Maths Society President 1971 - 1972 AMS Steele Prize 1998 |
|
[原创]群论的一些基础知识
问下JacobianMatrix,JacobianIdeal里的Jacobian是不是和数论里那个Jacobi符号的人相同?还是群论的大师JacobSON? 问下y^2-x^3+6的除子群为何是Group of divisors of Curve over Rational Field defined by $.1^3 - $.2^2*$.3 - 6*$.3^3? JacobianMatrix,JacobianIdeal A<x,y> := AffineSpace(Rationals(),2); A; C := Curve(A,y^2-x^3-4*x^2-16+x^12-y); C; SingularPoints(C); Qi<i> := QuadraticField(-3); SingularPoints(C,Qi); AmbientSpace(C); BaseRing(C); CoefficientRing(C); DefiningPolynomial(C) ; DefiningIdeal(C) ; Degree(C); JacobianIdeal(C); JacobianMatrix(C); Genus(C)ï¼? Affine Space of dimension 2 Variables: x, y Curve over Rational Field defined by x^12 - x^3 - 4*x^2 + y^2 - y - 16 {@ @} {@ @} Affine Space of dimension 2 Variables: x, y Rational Field Rational Field x^12 - x^3 - 4*x^2 + y^2 - y - 16 Ideal of Polynomial ring of rank 2 over Rational Field Order: Lexicographical Variables: x, y Basis: [ x^12 - x^3 - 4*x^2 + y^2 - y - 16 ] 12 Ideal of Polynomial ring of rank 2 over Rational Field Order: Lexicographical Variables: x, y Basis: [ x^12 - x^3 - 4*x^2 + y^2 - y - 16, 12*x^11 - 3*x^2 - 8*x, 2*y - 1 ] Mapping from: Ideal of Polynomial ring of rank 2 over Rational Field to Polynomial ring of rank 2 over Rational Field [12*x^11 - 3*x^2 - 8*x 2*y - 1] 5 |
|
[原创]群论的一些基础知识
Z := IntegerRing(5); M := RModule(Z, 6); M; R:=RSpace(Z, 7); R; RM:=RMatrixSpace(Z, 2, 3) ; RM; Rank(M) ; Basis(M); Rank(RM) ; Basis(RM); Zero(M); Zero(RM); u:=Random(M); u; u1:=Random(RM); u1; ElementToSequence(u); v:=Random(M); v; v1:=Random(M); v1; u+v; u+v1; - u ; -v1; u[4]; u[3]; u[2]; u[1]; u[5]; Normalize(u); Support(u) ; Weight(u) ; Normalise(u); M ! 0 ; sub<M | [3],[ 0],[ 0 ],[0 ],[0],[ 0]>; u in M ; RModule(IntegerRing(5), 6) Full Vector space of degree 7 over IntegerRing(5) Full KMatrixSpace of 2 by 3 matrices over IntegerRing(5) 6 [ M: (1 0 0 0 0 0), M: (0 1 0 0 0 0), M: (0 0 1 0 0 0), M: (0 0 0 1 0 0), M: (0 0 0 0 1 0), M: (0 0 0 0 0 1) ] 6 [ [1 0 0] [0 0 0], [0 1 0] [0 0 0], [0 0 1] [0 0 0], [0 0 0] [1 0 0], [0 0 0] [0 1 0], [0 0 0] [0 0 1] ] M: (0 0 0 0 0 0) [0 0 0] [0 0 0] M: (4 1 0 1 2 1) [3 4 0] [0 3 4] [ 4, 1, 0, 1, 2, 1 ] M: (3 2 1 4 3 4) M: (4 0 3 3 1 3) M: (2 3 1 0 0 0) M: (3 1 3 4 3 4) M: (1 4 0 4 3 4) M: (1 0 2 2 4 2) 1 0 1 4 2 Normalise( g: (4 1 0 1 2 1) ) In file "/magma/package/Group/GrpMat/CompTree/GrpMat/util/basics.m", line 292, column 17: >> G := Generic(Parent(g)); ^ Runtime error in 'Generic': Bad argument types Argument types given: ModED { 1, 2, 4, 5, 6 } 5 Normalise( g: (4 1 0 1 2 1) ) In file "/magma/package/Group/GrpMat/CompTree/GrpMat/util/basics.m", line 292, column 17: >> G := Generic(Parent(g)); ^ Runtime error in 'Generic': Bad argument types Argument types given: ModED M: (0 0 0 0 0 0) >> sub<M | [3],[ 0],[ 0 ],[0 ],[0],[ 0]>; ^ Runtime error in sub< ... >: Rhs argument 1 is invalid for this constructor true |
|
[原创]群论的一些基础知识
同态符号立起来是圈积符号,看那印度人网站,群论真是太庞大! http://groupprops.subwiki.org/wiki/External_wreath_product |
|
[原创]群论的一些基础知识
哪个才是S4到Aut(D)自同态? s4:=Sym(3); s4; D := DirectProduct(s4, s4); s44:=s4*s4; s44; s2:=Sym(2); s2; w1:=WreathProduct(D,s2); w1; Order(w1); A:=AutomorphismGroup(D); A; hom< s4 -> A | (1, 2, 3) -> ((1, 2,3),(1, 2) -> (1, 2), (4, 5, 6) -> (4, 6, 5),(4, 5) -> (4, 5))>; Symmetric group s4 acting on a set of cardinality 3 Order = 6 = 2 * 3 { (1, 3, 2), (2, 3), (1, 3), (1, 2, 3), (1, 2), Id(s4) } Symmetric group s2 acting on a set of cardinality 2 Order = 2 Permutation group w1 acting on a set of cardinality 12 Order = 2592 = 2^5 * 3^4 (1, 7)(2, 8)(3, 9)(4, 10)(5, 11)(6, 12) (1, 2, 3) (1, 2) (4, 5, 6) (4, 5) 2592 A group of automorphisms of GrpPerm: D, Degree 6, Order 2^2 * 3^2 Generators: Automorphism of GrpPerm: D, Degree 6, Order 2^2 * 3^2 which maps: (1, 2, 3) |--> (1, 2, 3) (1, 2) |--> (1, 2) (4, 5, 6) |--> (4, 6, 5) (4, 5) |--> (4, 5) Automorphism of GrpPerm: D, Degree 6, Order 2^2 * 3^2 which maps: (1, 2, 3) |--> (1, 3, 2) (1, 2) |--> (1, 2) (4, 5, 6) |--> (4, 5, 6) (4, 5) |--> (4, 5) Automorphism of GrpPerm: D, Degree 6, Order 2^2 * 3^2 which maps: (1, 2, 3) |--> (1, 2, 3) (1, 2) |--> (2, 3) (4, 5, 6) |--> (4, 5, 6) (4, 5) |--> (4, 5) Automorphism of GrpPerm: D, Degree 6, Order 2^2 * 3^2 which maps: (1, 2, 3) |--> (1, 2, 3) (1, 2) |--> (1, 2) (4, 5, 6) |--> (4, 5, 6) (4, 5) |--> (5, 6) Automorphism of GrpPerm: D, Degree 6, Order 2^2 * 3^2 which maps: (1, 2, 3) |--> (4, 5, 6) (1, 2) |--> (4, 5) (4, 5, 6) |--> (1, 2, 3) (4, 5) |--> (1, 2) >> hom< s4 -> A | (1, 2, 3) -> ((1, 2,3),(1, 2) -> (1, 2), (4, 5, 6) -> (4, 6, ^ Runtime error in elt< ... >: No permutation group context in which to create cycle >> hom< s4 -> A | (1, 2, 3) -> ((1, 3,2),(1, 2) -> (1, 2), (4, 5, 6) -> (4, 6, ^ Runtime error in elt< ... >: No permutation group context in which to create cycle >> hom< s4 -> A | (1, 2, 3) -> ((1, 3,2),(1, 2) -> (2, 3), (4, 5, 6) -> (4, 5, ^ Runtime error in elt< ... >: No permutation group context in which to create cycle >> hom< s4 -> A | (1, 2, 3) -> ((1, 3,2),(1, 2) -> (2, 3), (4, 5, 6) -> (4, 5, ^ Runtime error in elt< ... >: No permutation group context in which to create cycle >> hom< s4 -> A | (1, 2, 3) -> ((4,5,6),(1, 2) -> (4,5), (4, 5, 6) -> (1,2,3), ^ Runtime error in elt< ... >: No permutation group context in which to create cycle |
|
[原创]群论的一些基础知识
直幂是不是直和?圈积好像和半直和有关 |
|
[分享]少林和尚在度假村
再来几张啊啊啊 |
|
[原创]群论的一些基础知识
问下什麽是圈积? 圈积后阶怎麽会那麽大? http://groupprops.subwiki.org/wiki/Center 设G是群,L(G)是G的所有子群的集合。即 L(G)={H|H≤G}, 对任意的H1,H2∈L(G),H1∩H2也是G的子群,而<H1∪H2>是由H1∪H2生成的子群(即包含着H1∪H2的最小的子群).在L(G)上定义包含关系,则L(G)关于包含关系构成一个格,称为G的子群格。易见在L(G)中,H1∧H2就是H1∩H2,H1∨H2就是<H1∪H2>. s4:=Sym(4); s4; s4s:=sub<s4 | (1,4)(2,3)> ; s4s; s4ss:=sub<s4 | (1,4)(2,3),(1,3)(2,4)> ; s4ss; Center (s4); Center (s4s); Center (s4ss); WreathProduct(s4ss,s4s); Centralizer(s4s, s4ss) ; Orbits(s4); Orbits(s4s); Orbits(s4ss); Stabilizer(s4,1); Stabilizer(s4,2); Stabilizer(s4,3); Stabilizer(s4,4); a:=s4!(1,4)(2,3); a; set:=GSet(s4); set; b:=set!3; b; i:=Identity(s4); i; c:=a/i; c; d:=Action(s4,set); d; s4:=Sym(4); s4; s4s:=sub<s4 | (1,4)(2,3)> ; s4s; s4ss:=sub<s4 | (1,4)(2,3),(1,3)(2,4)> ; s4ss; w1:=WreathProduct(s4ss,s4s); w1; w2:=WreathProduct(s4s,s4ss); w2; pw1:=PrimitiveWreathProduct(s4ss, s4s); pw1; pw2:=PrimitiveWreathProduct(s4s, s4ss); pw2; Order(w1); Order(w2); Order(w1); Order(pw1); Order(pw2); Order(s4); Order(s4s); Order(s4ss); Symmetric group s4 acting on a set of cardinality 4 Order = 24 = 2^3 * 3 Permutation group s4s acting on a set of cardinality 4 (1, 4)(2, 3) Permutation group s4ss acting on a set of cardinality 4 (1, 4)(2, 3) (1, 3)(2, 4) Permutation group w1 acting on a set of cardinality 16 (1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12) (1, 4)(2, 3) (5, 8)(6, 7) (1, 3)(2, 4) (5, 7)(6, 8) Permutation group w2 acting on a set of cardinality 16 (1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12) (1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16) (1, 4)(2, 3) Permutation group pw1 acting on a set of cardinality 256 Permutation group pw2 acting on a set of cardinality 256 512 64 512 512 64 24 2 4 |
|
[原创]群论的一些基础知识
123456 印度人的群站 http://groupprops.subwiki.org/wiki/Main_Page http://groupprops.subwiki.org/wiki/Symmetric_group:S3 |
|
[原创]群论的一些基础知识
万念俱灰这种成语只能出现在小说里。。。。。博士可算是名利场中的名。。。名利场可不是纸上考试( ⊙ o ⊙ )啊! 所以要淡定。。。冷静! 已经没问题了,因为只见书上有S3/A3={A3,(12)A3} 没S3/A3={A3,(13)A3}={A3,(23)A3}=S3/A3所以问的太牛角尖。其实我常装S4抽象群表和26字母的抽象群乘群表------别人装数独,常琢磨陪集,中心,中心化子,正规化子,共扼,置换化子,边缘化子都得用 其实我问的就是个置换群表到抽象群表之间抽象过程 http://d.wanfangdata.com.cn/periodical_gsgsxb201005002.aspx |
|
|
操作理由
RANk
{{ user_info.golds == '' ? 0 : user_info.golds }}
雪币
{{ experience }}
课程经验
{{ score }}
学习收益
{{study_duration_fmt}}
学习时长
基本信息
荣誉称号:
{{ honorary_title }}
能力排名:
No.{{ rank_num }}
等 级:
LV{{ rank_lv-100 }}
活跃值:
在线值:
浏览人数:{{ visits }}
最近活跃:{{ last_active_time }}
注册时间:{{ user_info.create_date_jsonfmt }}
勋章
兑换勋章
证书
证书查询 >
能力值