Homomorphisms are an important part of group theory, and Magma supports homomorphisms between groups. Many useful homomorphisms are returned by constructors and intrinsic functions. Examples of these are the quo constructor, the sub constructor and intrinsic functions such as OrbitAction and FPGroup, which are described in more detail elsewhere in this chapter. In this section we describe how the user may create their own homomorphisms with domain a matrix group.
Given the matrix group G, construct the homomorphism f : G to H given by the generator images in L. H must be a group. The clause L may be any one of the following types:Each image element specified by the list must belong to the same group H. In the cases where pairs are given the given elements of G must generate G.
- (a)
- A list of elements of H, giving images of the generators of G;
- (b)
- A list of pairs, where the first in the pair is an element of G and the second its image in H, where pairs may be given in either of the (equivalent) forms <g,h> or g -> h;
- (c)
- A sequence of elements of H, as in (a);
- (d)
- A set or sequence of pairs, as in (b);
The domain of the homomorphism f.
The codomain of the homomorphism f.
The image or range of the homomorphism f. This will be a subgroup of the codomain of f. The algorithm computes the image and kernel simultaneously (see [LGPS91]).
The kernel of the homomorphism f. This will be a normal subgroup of the domain of f. The algorithm computes the image and kernel simultaneously (see [LGPS91]).
Return the value true if the sequence Q defines a homomorphism from the group G to the group H. The sequence Q must have length Ngens(G) and must contain elements of H. The i-th element of Q is interpreted as the image of the i-th generator of G and the function decides if these images extend to a homomorphism. If so, the homomorphism is also returned.
> K<z> := CyclotomicField(20); > zz := RootOfUnity(10, K); > i := RootOfUnity(4, K); > cos := (zz+ComplexConjugate(zz))/2; > sin := (zz-ComplexConjugate(zz))/(2*i); > gl := GeneralLinearGroup(2, K); > M := sub< gl | [cos, sin, -sin, cos], [-1,0,0,1]>; > #M; 20 > S := SymmetricGroup(5); > f := hom<M->S |[S|(1,2,3,4,5), (1,5)(2,4)]>; > Codomain(f); Symmetric group S acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 > Image(f); Permutation group acting on a set of cardinality 5 Order = 10 = 2 * 5 (1, 2, 3, 4, 5) (1, 5)(2, 4) > Kernel(f); MatrixGroup(2, K) of order 2 Generators: [-1 0] [ 0 -1]
Given two matrix groups G and H of degrees m and n respectively, construct the direct product of G and H as a matrix group of degree m + n.
Given a sequence Q of n matrix groups, construct the direct product Q[1] x Q[2] x ... x Q[n] as a matrix group of degree equal to the sum of the degrees of the groups Q[i], (i = 1, ..., n).
Given a matrix group G over the finite field K and a subfield S of K, construct the semilinear extension of G over the subfield S.
Given a matrix group G and a permutation group H, construct action of the wreath product on the tensor power of G by H, which is the (image of) the wreath product in its action on the tensor power (of the space that G acts on). The degree of the new group is dk where d is the degree of G and k is the degree of H.
Given a matrix group G and a permutation group H, construct the wreath product G wreath H of G and H.
> K<w> := FiniteField(4); > G := SpecialUnitaryGroup(3, K); > D := DirectProduct(G, G); > D; MatrixGroup(6, GF(2, 2)) Generators: [ 1 w w 0 0 0] [ 0 1 w^2 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] [w 1 1 0 0 0] [1 1 0 0 0 0] [1 0 0 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] [ 1 0 0 0 0 0] [ 0 1 0 0 0 0] [ 0 0 1 0 0 0] [ 0 0 0 1 w w] [ 0 0 0 0 1 w^2] [ 0 0 0 0 0 1] [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 w 1 1] [0 0 0 1 1 0] [0 0 0 1 0 0] > Order(D); 46656 > H := SymmetricGroup(3); > E := WreathProduct(G, H); > Degree(E); 9 > Order(E); 60466176 > F := TensorWreathProduct(G, H); > Degree(F); 27 > Order(F); 6718464