Homomorphisms are a central concept in group theory, and Magma provides extensive facilities for group homomorphisms. 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, BlocksAction, FPGroup and RadicalQuotient, 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 permutation group.
Given the permutation 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;
- (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. The algorithm employed is described in [LGPS91].
> G := PermutationGroup< 12 | (1,6,7)(2,5,8,3,4,9)(11,12),
> (1,3)(4,9,12)(5,8,10,6,7,11) >;
> #G;
648
> x := G ! (1, 2, 3)(7, 8, 9)(10, 11, 12);
> x_class := {@ x ^ y : y in G @};
> #x_class;
8
> S := SymmetricGroup(8);
> images := [S![Index(x_class, x_class[i]^(G.j)):i in [1..8]] :j in [1..2]];
> f := hom< G -> S | images>;
The map f is the homomorphism of G onto the group induced by the action of the element x. We computer the images of some elements and then find the image and kernel of f.
> (G.1*G.-2) @ f; (2, 5, 7)(3, 8, 6) > ((G.1) @ f) * ((G.2) @ f) ^ -1; (2, 5, 7)(3, 8, 6) > H := Image(f); > H; Permutation group acting on a set of cardinality 8 Order = 24 = 2^3 * 3 (1, 2, 3, 4, 6, 5)(7, 8) (1, 2, 8, 4, 6, 7)(3, 5) > Kernel(f); Permutation group acting on a set of cardinality 12 Order = 27 = 3^3 (1, 2, 3)(4, 6, 5)(7, 8, 9)(10, 12, 11) (4, 5, 6)(7, 9, 8) (7, 9, 8)(10, 11, 12)
We now find the preimage of O2(H) as a subgroup of G.
> pCore(H, 2) @@ f; Permutation group acting on a set of cardinality 12 Order = 216 = 2^3 * 3^3 (4, 5, 6)(7, 9, 8) (1, 2, 3)(4, 6, 5)(7, 8, 9)(10, 12, 11) (1, 4, 2, 5, 3, 6)(7, 12, 9, 11, 8, 10) (1, 10, 3, 11, 2, 12)(4, 9, 5, 8, 6, 7) (2, 3)(4, 5)(8, 9)(11, 12) (7, 9, 8)(10, 11, 12)