The automorphism group of a finite matrix group may be computed in Magma, subject to the same restrictions on the group as when computing maximal subgroups. (That is, all of the non-abelian composition factors of the group must appear in a certain database.) The methods used are those described in Cannon and Holt [CH03]. The existence of an isomorphism between a given matrix group and any other type of finite group (GrpPerm or GrpPC) may also be determined using similar methods.
Given a finite matrix group G, construct the full automorphism group F of G. The function returns the full automorphism group of G as a group of mappings (i.e., as a group of type GrpAuto). The automorphism group F is also computed as a finitely presented group and can be accessed via the function FPGroup(F). A function PermutationRepresentation is provided that when applied to F, attempts to construct a faithful permutation representation of reasonable degree. The algorithm described in Cannon and Holt [CH03] is used.SmallOuterAutGroup: RngIntElt Default: 20000SmallOuterAutGroup := t: Specify the strategy for the backtrack search when testing an automorphism for lifting to the next layer. If the outer automorphism group O at the previous level has order at most t, then the regular representation of O is used, otherwise the program tries to find a smaller degree permutation representation of O.Print: RngIntElt Default: 0The level of verbose printing. The possible values are 0, 1, 2 or 3.PrintSearchCount: RngIntElt Default: 1000PrintSearchCount := s: If Print := 3, then a message is printed at each s-th iteration during the backtrack search for lifting automorphisms.Further information about the construction of the automorphism group and a description of machinery for computing with group automorphisms may be found in Chapter AUTOMORPHISM GROUPS.
> k<w> := GF(4);
> G := MatrixGroup< 3, k |
> [w^2, 0, 0, 0, w^2, 0, 0, 0, w^2],
> [w^2, 0, w^2, 0, w^2, w^2, 0, 0, w^2],
> [1, 0, 0, 1, 0, w, w^2, w^2, 0],
> [w, 0, 0, w^2, 1, w^2, w, w, 0],
> [w, 0, 0, 0, w, 0, 0, 0, w] >;
> G;
MatrixGroup(3, GF(2^2))
Generators:
[w^2 0 0]
[ 0 w^2 0]
[ 0 0 w^2]
[w^2 0 w^2]
[ 0 w^2 w^2]
[ 0 0 w^2]
[ 1 0 0]
[ 1 0 w]
[w^2 w^2 0]
[ w 0 0]
[w^2 1 w^2]
[ w w 0]
[ w 0 0]
[ 0 w 0]
[ 0 0 w]
> #G;
576
> A := AutomorphismGroup(G);
> #A;
3456
> OuterOrder(A);
72
> F := FPGroup(A);
> P := DegreeReduction(CosetImage(F, sub<F|>));
> P;
Permutation group P acting on a set of cardinality 48
Thus, we see that G has an automorphism group of order 3456 and the quotient group of A consisting of outer automorphisms, has order 72. The automorphism group may be realised as a permutation group of degree 48.
Test whether or not the two finite groups G and H are isomorphic as abstract groups. If so, both the result true and an isomorphism from G to H is returned. If not, the result false is returned. The algorithm described in Cannon and Holt [CH03] is used.SmallOuterAutGroup: RngIntElt Default: 20000SmallOuterAutGroup := t: Specify the strategy for the backtrack search when testing an automorphism for lifting to the next layer. If the outer automorphism group O at the previous level has order at most t, then the regular representation of O is used, otherwise the program tries to find a smaller degree permutation representation of O.Print: RngIntElt Default: 0The level of verbose printing. The possible values are 0, 1, 2 or 3.PrintSearchCount: RngIntElt Default: 1000PrintSearchCount := s: If Print := 3, then a message is printed at each s-th iteration during the backtrack search for lifting automorphisms.
> n := 4;
> N := 4*n;
> K<z> := CyclotomicField(N);
> zz := z^4;
> i := z^n;
> cos := (zz+ComplexConjugate(zz))/2;
> sin := (zz-ComplexConjugate(zz))/(2*i);
> gl := GeneralLinearGroup(3, K);
> G := sub< gl | [ cos, sin, 0,
> -sin, cos, 0,
> 0, 0, 1 ],
>
> [ -1, 0, 0,
> 0, 1, 0,
> 0, 0, 1 ] >;
>
> #G;
8
> D8 := DihedralGroup(4);
> D8;
Permutation group G acting on a set of cardinality 4
Order = 8 = 2^3
(1, 2, 3, 4)
(1, 4)(2, 3)
> #D8;
8
> bool, iso := IsIsomorphic(G, D8);
> bool;
true
> iso;
Homomorphism of MatrixGroup(3, K) of order 2^3 into
GrpPerm: D8, Degree 4, Order 2^3 induced by
[ 0 1 0]
[-1 0 0]
[ 0 0 1] |--> (1, 2, 3, 4)
[-1 0 0]
[ 0 1 0]
[ 0 0 1] |--> (1, 3)