Let G be a matrix group and let M be its natural module. Now G has an action on the elements and submodules of M. A derived G-set for G consists of the closure under the natural action of G of one of the following:
Given an element g belonging to the matrix group G with natural module M and an element u of this module, return the vector u * g.
Given an element g belonging to the matrix group G with natural module M and an object y which is an element of some derived G-set of M, find the image of y under g.
Given a matrix group G with natural module M and an object y which is either a vector of M, a submodule of M, or a tuple whose components are either vectors or submodules, find the orbit of y under G.
Given a matrix group G with natural module M and an object y which is either a vector of M, a submodule of M, or a tuple whose components are either vectors or submodules, return true if the orbit of y under G has length less than or equal to b. Otherwise the function returns false. If it returns true, then the orbit of y is returned as the second value.
Given a matrix group G with natural R-module M, construct the orbits of G on the vectors of M. The orbits are returned as a sequence of sets.
Given a matrix group G with natural R-module M, construct the orbits of G on the rank-1 submodules of M. The orbits are returned as a sequence of sets.
Given a matrix group G with natural module M and a set S of vectors or subspaces of M, return the union of orbits of the elements of S under the natural action of G on M.
Given a matrix group G with natural module M and an object y which is either a vector of M, a submodule of M, or a tuple whose components are either vectors or submodules, determine the stabilizer of y in G.
> V := RSpace(G); > u := V![1,0,0,0,0,0]; > U := sub< V | u >; > x := < u, U >; > W := sub< V | u, u*G.1 >; > u^G.1; (w^6 w^3 w^2 0 0 0) > U^G.1; Vector space of degree 6, dimension 1 over GF(3, 2) Echelonized basis: ( 1 w^5 2 0 0 0) > W^G.1; Vector space of degree 6, dimension 2 over GF(3, 2) Echelonized basis: ( 1 w^5 0 0 0 0) ( 0 0 1 0 0 0) > x^G.1; <(w^6 w^3 w^2 0 0 0), Vector space of degree 6, dimension 1 over GF(3, 2) Echelonized basis: ( 1 w^5 2 0 0 0)> > H := sub< G | G.1, G.2 >; > #Orbit(H, u); 252 > #Orbit(H, U); 63 > #Orbit(G, U); 3150 > Stabilizer(G, U); MatrixGroup(6, GF(3^2)) of order 384 = 2^7 * 3 Generators: [ 2 0 0 0 0 0] [w^3 w w 0 2 w^2] [w^5 w^7 w^7 0 1 w^2] [ 0 0 1 2 1 0] [w^7 w^5 0 0 0 w^6] [ w w^3 0 0 0 w^6]
[w^2 0 0 0 0 0] [w^5 w^5 w^5 0 w 0] [w^7 w^3 w^3 0 0 w^7] [w^2 w^3 w w^6 w w^3] [w^3 1 w^6 0 w w^7] [ w w^6 2 0 w w^7]
[w^6 0 0 0 0 0] [ 0 2 0 0 0 0] [ 0 0 w^6 0 0 0] [w^2 w^7 w^6 w^2 0 0] [ w 0 w 0 2 0] [w^6 w^7 w^2 0 0 w^2]
[ 2 0 0 0 0 0] [ 0 2 0 0 0 0] [ 0 0 2 0 0 0] [ 0 0 0 2 0 0] [ 0 0 0 0 2 0] [ 0 0 0 0 0 2] > #Orbit(H, x); 252 > #Orbit(H, W); 28
Given a matrix group G with natural module M, and a set T consisting of either (a) elements of M, (b) submodules of M or (c) tuples, form the G-closure Y of T and construct the homomorphism phi: G -> L, where the permutation group L gives the action of G on the set Y. The function returns:
- The natural homomorphism phi: G -> L;
- The induced group L;
- The kernel of the action (a subgroup of G).
Given a matrix group G with natural module M, and a set T consisting of either (a) elements of M, (b) submodules of M or (c) tuples, form the G-closure Y of T. If the cardinality of Y does not exceed b, then construct the homomorphism phi: G -> L, where the permutation group L gives the action of G on the set Y. In this case the function returns:
- The boolean value true.
- The natural homomorphism phi: G -> L;
- The induced group L;
- The kernel of the action (a subgroup of G). If the cardinality of Y exceeds b, simply return false. (The action of G on Y is not constructed in this case).
Given a matrix group G with natural module M, and a set T consisting of either (a) elements of M, (b) submodules of M or (c) tuples, form the G-closure Y of T and return the permutation group L giving the action of G on Y.
Given a matrix group G with natural module M, and set T consisting of either (a) elements of M, (b) submodules of M or (c) tuples, form the G-closure Y of T. If the cardinality of Y does not exceed b, return true togther with the permutation group L giving the action of G on Y. If the cardinality of Y does exceed b, the action is not constructed and the single value false is returned.
Given a matrix group G with natural module M, and a set T consisting of either (a) elements of M, (b) submodules of M or (c) tuples, form the G-closure Y of T and return the the permutation group L giving the action of G on Y.
Given a matrix group G with natural module M, and set T consisting of either (a) elements of M, (b) submodules of M or (c) tuples, form the G-closure Y of T. If the cardinality of Y does not exceed b, return the boolean value true together with the kernel of the action of G on Y. If the cardinality of Y does exceed b, the kernel is not constructed and the single value false is returned.
> [ Factorization(CharacteristicPolynomial(G.i)) : i in [1..3] ];
[
[
<x^3 + w^5*x^2 + w^3*x + 2, 1>,
<x^3 + w^7*x^2 + w*x + 2, 1>
],
[
<x + 2, 6>
],
[
<x + w^2, 3>,
<x + w^6, 3>
]
]
> y := Eigenspace(G.2, -2);
> y;
Vector space of degree 6, dimension 3 over GF(3, 2)
Echelonized basis:
(1 0 0 1 2 1)
(0 1 0 2 1 2)
(0 0 1 1 2 1)
> #Orbit(G, y);
280
> P := OrbitImage(G, y);
> P;
Permutation group P of degree 280
> Order(P);
604800
> CompositionFactors(P);
G
| J2
1
> // Thus, our group has the simple group J_2 of Janko as a composition factor.
> Order(G);
1209600
> //Hence the kernel of this action has order 2.
Right coset of the subgroup H of the group G, where g is an element of G.
True if element g of group G lies in the coset C.
True if element g of group G does not lie in the coset C.
True if the coset C_1 is equal to the coset C_2.
True if the coset C_1 is not equal to the coset C_2.
The cardinality of the coset C.
The (right) coset table for G over subgroup H relative to its defining generators.
The coset table for G corresponding to the permutation representation f of G, where f is a homomorphism of G onto a transitive permutation group.
Given a matrix group G and a subgroup H of G, this function returns
- A set of elements T of G forming a right transversal for G over H; and
- The corresponding transversal mapping phi: G -> T. If T = [t_1, ..., t_r] and g in G, phi is defined by phi(g) = t_i, where g in H * t_i.
Given a subgroup H of the group G, construct the permutation representation of G given by the action of G on the (right) coset space cos(G, H). The function returns:
- The natural homomorphism f: G -> L;
- The induced permutation group L;
- The kernel K of the action (a subgroup of G).
Given a subgroup H of the group G, construct the image L of G given by the action of G on the (right) coset space cos(G, H). L is returned as a permutation group.
Given a subgroup H of the group G, construct the kernel of the action of G on the (right) coset space cos(G, H).
> G := MatrixGroup< 3, GF(3) | [0,2,0, 1,1,0, 0,0,1], [0,1,0, 0,0,1, 1,0,0] >;
> Order(G);
5616
> H := sub< G | G.1^2, G.2 >;
> Order(H);
216
> P := CosetImage(G, H);
> P;
Permutation group P of degree 26
(1, 2)(3, 4, 6, 5, 7, 9)(8, 11)(10, 13, 15, 20, 18, 17)
(12, 16, 21, 14, 19, 24)(23, 26)
(2, 3, 5)(4, 6, 8)(7, 10, 14)(9, 12, 17)(11, 15, 20)(13, 18, 23)
(16, 22, 21)(19, 25, 24)