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
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
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]
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
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
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)
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)
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
