In this section the following topics are covered:
The specialisation of the general hom--constructor to the case in which the domain is an fp-group is described in this section.
Constructor/Intrinsic:
hom< P -> G | S > : Returns the homomorphism from the n-generator fp-group P to the group G defined by the assignment S. Here S is either a list, sequence or indexed set containing the images of the n generators of P or a list, sequence, enumerated set or indexed set, containing n tuples < xi, yi > or n arrow pairs xi -> yi defining the n generator-image pairs.
IsSatisfied(U, E) : Let U be a set or sequence of words or relations constituting the defining presentation of an n-generator fp-group H and E be a sequence of n elements in a group G for which there is a unique representation for each element. This intrinsic evaluates the words/relations of U in G where the i-th generator of H maps to the i-th element of G and returns true if and only if each word in U evaluates to the identity or each relation in U is satisfied.
Notes:
The first step is to create the braid group B on 5 strings, that is, 4 Artin generators.
> B<x,y,z,t> := BraidFPGroup(5);
> B;
Finitely presented group B on 4 generators
Relations
x * y * x = y * x * y
x * z = z * x
x * t = t * x
y * z * y = z * y * z
y * t = t * y
z * t * z = t * z * t
In the symmetric group of degree 5, 4 transpositions are defined which will be
the images of the generators of B.
> S := Sym(5); > imgs := [ S!(1,2), S!(2,3), S!(3,4), S!(4,5) ];In order to verify that this assignment actually gives rise to a well-defined homomorphism, a check is made to decide whether the potential images satisfy the defining relations of B.
> rels := Relations(B); > rels; [ x*y*x = y*x*y, x*z = z*x, x*t = t*x, y*z*y = z*y*z, y*t = t*y, z*t*z = t*z*t ] > IsSatisfied(rels, imgs); trueThey do. So now the homomorphism from B to S can be defined:-
> f := hom< B->S | imgs >;Note that f is surjective, i.e., S is an epimorphic image of B as claimed above.
> f(B) eq S; trueWhat is the kernel of f?
> Kernel(f); Finitely presented group Index in group B is 120 = 2^3 * 3 * 5 Subgroup of group B defined by coset tableUsing the function GeneratingWords described elsewhere, a set of generators for ker(f) as a subgroup of B can be constructed.
> GeneratingWords(B, Kernel(f));
{ x^-2, y^-2, z^-2, t^-2, (y*z^-1*y^-1)^2, (z*t^-1*z^-1)^2,
(x*y^-1*x^-1)^2, (x*y*z^-1*y^-1*x^-1)^2, (y*z*t^-1*z^-1*y^-1)^2,
(x*y*z*t^-1*z^-1*y^-1*x^-1)^2 }
It is easy to see that all generators of ker(f) are conjugates of words of
the form gpm2, where g is a generator of B. This is checked, using the
normal closure intrinsic NormalClosure.
> N := sub< B | x^2, y^2, z^2, t^2 >; > Kernel(f) eq NormalClosure(B, N); trueThus the braid relations together with the relators x2, y2, z2, t2 are a set of defining relations for S. This is confirmed by running the following code:-
> BS := quo< B | x^2, y^2, z^2, t^2 >; > Order(BS); 120 > P := PermutationGroup(BS); > P; Permutation group P acting on a set of cardinality 10 Order = 120 = 2^3 * 3 * 5 > IsIsomorphic(P, Sym(5)); true
This section describes functions for computing representatives of the classes of homomorphisms from a finitely presented group G to a finite group H modulo a group A of automorphisms of H. The algorithm and code are due to Derek Holt.
Intrinsics:
Homomorphisms (G, H, A ) : Given a finitely presented group G and two permutation groups H and A with H triangleleft A, a sequence containing representatives of the classes of homomorphisms from G to H modulo automorphisms of G induced by elements of A is returned.
Homomorphisms (G, H, A ) : This version is identical except that H and A are polycyclic groups rather than permutation groups.
Notes:
> G := FPGroup< a,b,c | a*c^-1*b*c^-1*a*b*a^-1*b, > a*b*a*b^-1*c^2*b^-1, > a^2*b^-1*c*a*c*a*c*a*c*a*c*b^-1 >;The intrinsic Homomorphisms will be used to prove that there is a homomorphism from G onto A5.
> H := Alt(5);
> homs := Homomorphisms(G, H : Limit := 1);
> homs;
[
Homomorphism of GrpFP: G into GrpPerm: H, Degree 5, Order 2^2 * 3 * 5
G.1 |--> (1, 2, 3)
G.2 |--> (1, 2, 4)
G.3 |--> (2, 4)(3, 5)
]
Hence there exists at least one such homomorphism. Are there any more?
> homs := Homomorphisms(G, H); > #homs; 4So there are four distinct classes of homomorphisms! Finally, the kernel of the first of these homomorphisms is found.
> phi := homs[1]; > K := Kernel(phi); > K; Finitely presented group K Index in group G is 60 = 2^2 * 3 * 5 Subgroup of group G defined by coset table
This section describes an intrinsic that searches for isomorphisms between two finitely presented groups. The algorithm and code are due to Derek Holt.
Intrinsics:
SearchForIsomorphism (G, H, m) : Attempt to find an isomorphism from the finitely presented group G to the finitely presented group H. The search will be restricted to those homomorphisms for which the sum of the word-lengths of the images of the generators of G in H is at most m. If an isomorphism φ is found, then the values true, φ, φ - 1 are returned. Otherwise, the values false, _, _ are returned; of course, that does not necessarily mean that the groups are not isomorphic.
Notes:
> G1<s,t,u> := FPGroup <s,t,u | s*u*s^-1=u^-1, t^2=u^2, t*s^2*t^-1=s^-2,
> u*(s*t)^2=(s*t)^2*u >;
> G2<x,y,z> := FPGroup<x,y,z | x*y^2*x^-1=y^-2, y*x^2*y^-1=x^-2, x^2=z^2*(x*y)^2,
> y^2=(z^-1*x)^2, z*(x*y)^2=(x*y)^2*z >;
> isiso, f1, f2 := SearchForIsomorphism(G1,G2,7);
> isiso;
true
> f1;
Homomorphism of GrpFP: G1 into GrpFP: G2 induced by
s |--> x * z^-1
t |--> y * z
u |--> x * y^-1 * z
> f2;
Homomorphism of GrpFP: G2 into GrpFP: G1 induced by
x |--> s^2 * u * t^-1
y |--> s^-1 * u^-1
z |--> s * u * t^-1
So the search for an isomorphism succeeded and returned the isomorphism and its
inverse.
> G1<x,y,z> := FPGroup< x,y,z | x^2*y^5, x^14*z^23, (x^2,y), (x^2,z), x*y*z>; > G2<a,b> := FPGroup<a,b | a*b^16*a*b^-7, a^4*b^7*a^-1*b^7>;It is a good idea to minimize the number of generators of G1 since there is clearly a redundant generator.
> G1s := Simplify(G1); > Ngens(G1s); 2 > G1!G1s.1, G1!G1s.2; x zNow SearchForIsomorphism (G1, G2, 15) will succeed, but will take many hours, and also use a lot of memory. In contrast to G1, it can sometimes help to introduce a new generator in the presentation for G2 if there are common substrings in relators.
> G2b<a,b,c> := FPGroup< a,b,c | a*b^16*a*b^-7, a^4*b^7*a^-1*b^7, c=b^7>;
> isiso, f1, f2 := SearchForIsomorphism(G1s,G2b,4);
> isiso;
true
> f1;
Homomorphism of GrpFP: G1s into GrpFP: G2b induced by
G1s.1 |--> a * c^-1
G1s.2 |--> c
> f2;
Homomorphism of GrpFP: G2b into GrpFP: G1s induced by
a |--> G1s.1 * G1s.2
b |--> G1s.1^6 * G1s.2^10
c |--> G1s.2