To compute with automorphism groups, Magma uses various concrete representations of the group. These are summarised in this section.
Construct a permutation representation of the group of automorphisms A. The function finds a union of conjugacy classes of the base group G which is closed under the action of A and with G-normal closure equal to G. The permutation action of A on such a set is faithful. The results returned are the representation of A as a homomorphism A to P, the image of this homomorphism as a permutation group with standard support, and the set of elements of G used.
Given a group of automorphisms A of a group G, this function returns a permutation group isomorphic to A as defined in the description of the function PermutationRepresentation.
Given a group of automorphisms A of a group G, this function returns the set of elements of G (i.e., a union of conjugacy classes) used as the support of the permutation group constructed by the PermutationRepresentation function.
Attempt to directly construct a pc-representation for the group of automorphisms A of a conditioned p-group G. A must have been constructing using the AutomorphismGroup intrinsic, or any equivalent alias. The results returned are a boolean value indicating the solubility of A, and if soluble, a representation of A as a homomorphism A to P and the image of this homomorphism as a pc-group.
A presentation for the group of automorphisms A on the generators of A. The isomorphism from the finitely presented group to the group of automorphisms A is also returned.
Suppose that A is the full group of automorphisms of a group G. This function returns a finitely presented group O isomorphic to the outer automorphism group of the base group G. The natural homomorphism from FPGroup(A) onto O is also returned.
> G := PGL(2, 9);
> A := AutomorphismGroup(G);
> PermutationGroup(A);
Permutation group acting on a set of cardinality 36
Order = 1440 = 2^5 * 3^2 * 5
(1, 30)(3, 27)(5, 17)(6, 24)(8, 9)(10, 14)(11, 13)(12, 32) (15, 34)(16, 21)
(18, 25)(19, 28)(22, 29)(23, 31)(26, 33)(35, 36)
(1, 32, 19, 22)(2, 34)(3, 18, 7, 17)(4, 25, 30, 31) (5, 23, 33, 24)
(6, 15, 26, 21)(8, 16, 29, 20)(9, 35, 14, 27) (10, 13, 11, 12)(28, 36)
(1, 2, 3, 5, 8, 13, 22, 31)(4, 7, 9, 15, 24, 26, 34, 29)
(6, 10, 17, 23, 32, 33, 28, 35)(11, 19, 27, 16, 25, 21, 30, 36)
(12, 20, 14, 18)
(1, 32, 33, 12, 21, 6)(2, 34, 26, 35, 17, 16)(3, 28, 22, 7, 18, 13)
(4, 31, 20, 29, 24, 11)(5, 19, 25, 10, 15, 8)(9, 30, 36, 14, 23, 27)
> F<x, y, z, t> := FPGroup(A);
> F;
Finitely presented group F on 4 generators
Relations
x^2 = Id(F)
y^4 = Id(F)
(x * y^-1)^5 = Id(F)
y^-2 * x * y^-2 * x * y^-2 * x * y^2 * x * y^2 * x = Id(F)
z^-1 * x * z * y^-1 * x^-1 * y^-2 * x^-1 * y * x^-1 *
y^-2 * x^-1 * y * x^-1 * y^-1 * x^-1 = Id(F)
z^-1 * y * z * y * x^-1 * y * x^-1 * y^-1 * x^-1 = Id(F)
z^2 * y^-1 * x^-1 * y * x^-1 * y^-1 * x^-1 = Id(F)
x^t = y * x * y^-1
y^t = y^-1 * x * y * x * y
z^t = z * x * y^-1 * x
t^2 = x * y^2 * x * y^-1 * x * y
> load hs100; Loading "/home/magma/libs/pergps/hs100" The simple group of Higman-Sims represented as a permutation group of degree 100. Order: 44 352 000 = 2^9 * 3^2 * 5^3 * 7 * 11. Base: 1, 2, 3, 4, 5, 6. Group: G > aut := AutomorphismGroup(G); > P := PermutationGroup(aut); > P; Permutation group P acting on a set of cardinality 5775 Order = 88704000 = 2^10 * 3^2 * 5^3 * 7 * 11We've got a permutation representation on 5775 letters. Now we want to get it on 100 letters, so we need to find the subgroup of index 100.
> lix := LowIndexSubgroups(P, 100); > [ Index(P, H) : H in lix]; [ 1, 2, 100 ]There it is, so we can compute the corresponding permutation representation.
> H := CosetImage(P, lix[3]);
> H;
Permutation group H acting on a set of cardinality 100
> CompositionFactors(H);
G
| Cyclic(2)
*
| HS
1