In this section various intrinsics for constructing fp-groups are described. These are familiar groups constructed as fp-groups, extensions of existing fp-groups and the construction of fp-groups from some other type of group.
A number of functions are provided which construct presentations for various standard groups.
Construct the abelian group defined by the sequence [n1, ..., nr] of non-negative integers as an fp-group. The function returns the direct product of cyclic groups Cn1 x Cn2 x ... x Cnr, where C0 is interpreted as an infinite cyclic group.
Construct the alternating group of degree n as an fp-group, where the generators correspond to the permutations (3, 4, ... , n) and (1, 2, 3), for n odd, or (1, 2)(3, 4, ..., n) and (1, 2, 3), for n even.
Construct the braid group on n strings (n - 1 Artin generators) as an fp-group.
Construct the Coxeter group of Cartan type t as a finitely presented group, given by the standard Coxeter presentation. The Cartan type t is passed to this function as a string; we refer to Chapter ROOT DATA for details.
Construct the cyclic group of order n as an fp-group.
For n > 2, return the dihedral group of order 2n as an fp-group. For n=0, return the infinite dihedral group as an fp-group.
Given a small prime p and a small positive integer n, construct an extra-special group G of order p2n + 1 in the category GrpFP. The isomorphism type of G can be selected using the parameter Type.Type: MonStgElt Default: "+"Possible values for this parameter are "+" (default) and "-".If Type is set to "+", the function returns for p = 2 the central product of n copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p2n + 1 and exponent p.
If Type is set to "-", the function returns for p = 2 the central product of a quaternion group of order 8 and n - 1 copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p2n + 1 and exponent p2.
Construct the symmetric group of degree n as an fp-group, where the generators correspond to the permutations (1, 2, ..., n) and (1, 2).
> S8 := SymmetricFPGroup(8);
> S8;
Finitely presented group S8 on 2 generators
Relations
S8.1^8 = Id(S8)
S8.2^2 = Id(S8)
(S8.1 * S8.2)^7 = Id(S8)
(S8.1^-1 * S8.2 * S8.1 * S8.2)^3 = Id(S8)
(S8.2 * S8.1^-2 * S8.2 * S8.1^2)^2 = Id(S8)
(S8.2 * S8.1^-3 * S8.2 * S8.1^3)^2 = Id(S8)
(S8.2 * S8.1^-4 * S8.2 * S8.1^4)^2 = Id(S8)
We create the Coxeter group of Cartan type F4 as an fp-group:
> F := CoxeterFPGroup("F4");
> F;
Finitely presented group F on 4 generators
Relations
(F.2 * F.3)^2 = (F.3 * F.2)^2
F.1^2 = Id(F)
F.1 * F.3 = F.3 * F.1
F.2 * F.4 = F.4 * F.2
F.1 * F.2 * F.1 = F.2 * F.1 * F.2
F.2^2 = Id(F)
F.3^2 = Id(F)
F.3 * F.4 * F.3 = F.4 * F.3 * F.4
F.4^2 = Id(F)
F.1 * F.4 = F.4 * F.1
Given an fp-group G, construct a maximal central extension tilde G of G. The group tilde G is created as an fp-group.
Given two fp-groups G and H, construct the direct product of G and H.
Given a sequence Q of r fp-groups, construct the direct product Q[1] x ... x Q[r].
Given two fp-groups G and H, construct the free product of G and H.
Given a sequence Q of r fp-groups, construct the free product of the groups Q[1], ..., Q[r].
> G<x1, x2> := FPGroup<x1, x2 | x1^4,(x1*x2^-1)^2,x2^4,(x1*x2)^3>;
> G;
Finitely presented group G on 2 generators
Relations
x1^4 = Id(G)
(x1 * x2^-1)^2 = Id(G)
x2^4 = Id(G)
(x1 * x2)^3 = Id(G)
> D := Darstellungsgruppe(G);
> D;
Finitely presented group D on 4 generators
Relations
D.1^4 * D.3^-1 * D.4^2 = Id(D)
D.1 * D.2^-1 * D.1 * D.2^-1 * D.4 = Id(D)
D.2^4 = Id(D)
D.1 * D.2 * D.1 * D.2 * D.1 * D.2 * D.4 = Id(D)
(D.1, D.3) = Id(D)
(D.2, D.3) = Id(D)
(D.1, D.4) = Id(D)
(D.2, D.4) = Id(D)
(D.3, D.4) = Id(D)
> Index(D, sub< D | >);
108
Thus, a maximal central extension of G has order 108
> A5 := FPGroup<a, b | a^2, b^3, (a*b)^5 >;
> Z2 := quo< FreeGroup(1) | $.1^2 >;
> G := DirectProduct(A5, Z2);
> G;
Finitely presented group G on 3 generators
Relations
G.1^2 = Id(G)
G.2^3 = Id(G)
(G.1 * G.2)^5 = Id(G)
G.3^2 = Id(G)
G.1 * G.3 = G.3 * G.1
G.2 * G.3 = G.3 * G.2
In this section we describe intrinsics which given a group in some category other than that of fp-group, return an isomorphic fp-group.
Constructing a nice presentation for a large finite permutation group or matrix group is a difficult problem. The next two intrinsics obtain a presentation as a side effect of the Todd-Coxeter-Schreier-Sims algorithm.
It should be noted that FPGroupStrong constructs a presentation on what is often a much larger generating set and consequently the presentation can be much larger. However, this intrinsic will produce presentations for very large groups as compared to the intrinsic FPGroup. It is recommended that the user apply the simplification algorithm (intrinsic Simplify) to the presentation returned by FPGroupStrong).
Groups that satisfy certain properties, such as being abelian or polycyclic, are known to possess presentations with respect to which the word problem is soluble. Specialised categories have been constructed in Magma for several of these, e.g. the categories GrpGPC, GrpPC and GrpAb. The functions described in this section allow a group created in one of the special presentation categories to be recast as an fp-group.
There is a special Magma category GrpPermCox, a subcategory of GrpPerm, for finite Coxeter groups. Here, we describe a function to create from a Coxeter group W a finitely presented group F, isomorphic to W, which is given by the standard Coxeter group presentation. We refer to Chapter COXETER GROUPS for the details.
Given a finite group G in category GrpPerm or GrpMat, this function returns a finitely presented group F, isomorphic to G, together with the isomorphism φ: F -> G. The generators of F correspond to the generators of G, so this function can be used to obtain a set of defining relations for the given generating set of G.It should be noted that this function is only practical for groups of order at most a few million. In the case of much larger permutation groups, an isomorphic fp-group can be constructed using the function FPGroupStrong.
> G := Alt(10);
> G;
Permutation group G acting on a set of cardinality 5
Order = 1814400 = 2^7 * 3^4 * 5^2 * 7
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10)
(1, 2, 3)
Now we create an fp-group F isomorphic to G, using the function
FPGroup. The presentation is
constructed by computing a set of defining relations for the generators
of G, i.e. the generators of the returned fp-group correspond to
the generators of G. This defines a homomorphism from F to G,
which the function FPGroup returns as second return value.
> F<x,y>, f := FPGroup(G);
> F;
Finitely presented group F on 2 generators
Relations
x^8 = Id(F)
y^3 = Id(F)
(x^-1, y^-1)^2 = Id(F)
(x * y^-1 * x^-2 * y^-1 * x)^2 = Id(F)
(x * y * x^-3 * y^-1 * x^2)^2 = Id(F)
x^-1 * y^-1 * x^-4 * y^-1 * x^4 * y^-1 * x^4 * y^-1 * x^-3 = Id(F)
(y^-1 * x^-1)^9 = Id(F)
> f;
Mapping from: GrpFP: F to GrpPerm: G
> f(x);
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10)
> f(y);
(1, 2, 3)
Given a finite group G in category GrpPerm or GrpMat, this function returns a finitely presented group F, isomorphic to G, together with the isomorphism φ:F -> G. The generators of F correspond to a set of strong generators of G. If no strong generating set is known for G, one will be constructed.For a detailed description of this function, in particular for a list of available parameters, we refer to Chapter PERMUTATION GROUPS and Chapter MATRIX GROUPS OVER GENERAL RINGS, respectively.
Given a permutation group G and a normal subgroup N of G, this function returns a finitely presented group F, isomorphic to G/N, together with a homomorphism φ:G -> F.For a detailed description of this function, we refer to Chapter PERMUTATION GROUPS.
> G := Alt(5);
> G;
Permutation group G acting on a set of cardinality 5
Order = 60 = 2^2 * 3 * 5
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10)
(1, 2, 3)
Using the function FPGroupStrong, we now create an
fp-group Fs, isomorphic to G, whose generators correspond to a set of
strong generators of G.
> Fs<[z]>, fs := FPGroupStrong(G);
> Fs;
Finitely presented group Fs on 3 generators
Relations
z[1]^-3 = Id(Fs)
(z[1]^-1 * z[3]^-1)^2 = Id(Fs)
z[3]^-3 = Id(Fs)
z[1]^-1 * z[2]^-1 * z[1] * z[3]^-1 * z[2] = Id(Fs)
z[2]^-3 = Id(Fs)
(z[3]^-1 * z[2]^-1)^2 = Id(Fs)
> fs;
Mapping from: GrpFP: Fs to GrpPerm: G
Applying the isomorphism fs, we have a look at the strong generating set
constructed for G.
> [ fs(z[i]) : i in [1..#z] ];
[
(3, 4, 5),
(1, 2, 3),
(2, 3, 4)
]
Given a group G, defined either by a polycyclic group presentation (types GrpPC and GrpGPC) or an abelian group presentation (type GrpAb), return a group H isomorphic to G, together with the isomorphism φ:G -> H. The generators for H will correspond to the generators of G. The effect of this function is to convert a presentation in a special form into a general fp-group presentation.
> G := DihedralGroup(GrpGPC, 0);
> G;
GrpGPC : G of infinite order on 2 PC-generators
PC-Relations:
G.1^2 = Id(G),
G.2^G.1 = G.-2
> F := FPGroup(G);
> F;
Finitely presented group F on 2 generators
Relations
F.1^2 = Id(F)
F.2^F.1 = F.2^-1
F.1^-1 * F.2^-1 * F.1 = F.2
Given a finite Coxeter group W in the category GrpFPCox or
GrpPermCox, construct a
finitely presented group F isomorphic to W, given by a standard Coxeter
presentation. The isomorphism from W to F is returned as second return
value. The first argument to this function must be the category GrpFP.
Local: BoolElt Default: false
If the parameter Local is set to true, F is the appropriate
subgroup of the FP version of the overgroup of W.
> W := CoxeterGroup("C5");
> F<[s]>, h := CoxeterFPGroup(W);
> F;
Finitely presented group F on 5 generators
Relations
s[3]^2 = Id(F)
s[2] * s[4] = s[4] * s[2]
s[1]^2 = Id(F)
s[1] * s[2] * s[1] = s[2] * s[1] * s[2]
s[2] * s[5] = s[5] * s[2]
s[4]^2 = Id(F)
s[1] * s[3] = s[3] * s[1]
s[3] * s[4] * s[3] = s[4] * s[3] * s[4]
s[2]^2 = Id(F)
s[1] * s[4] = s[4] * s[1]
s[5]^2 = Id(F)
(s[4] * s[5])^2 = (s[5] * s[4])^2
s[3] * s[5] = s[5] * s[3]
s[1] * s[5] = s[5] * s[1]
s[2] * s[3] * s[2] = s[3] * s[2] * s[3]
> h;
Mapping from: GrpCox: W to GrpFP: F given by a rule
> h(W.1*W.2);
s[1] * s[2]
> (s[1]*s[2]*s[3]*s[4]) @@ h;
(1, 39, 4, 3, 2)(5, 13, 19, 23, 25)(6, 36, 35, 8, 7)(9, 16, 21,
24, 17)(10, 33, 32, 31, 11)(12, 18, 22, 15, 20)(14, 29, 28,
27, 26)(30, 38, 44, 48, 50)(34, 41, 46, 49, 42)(37, 43, 47,
40, 45)