Construct the free group F of rank n, where n is a positive integer.The i-th generator of F may be referenced by the expression F.i, i = 1, ..., n. Note that a special form of the assignment statement is provided which enables the user to assign names to the generators of F. In this form of assignment, the list of generator names is enclosed within angle brackets and appended to the variable name on the left hand side of the assignment statement: F< v_1, ..., v_n > := FreeGroup(n);
> F := FreeGroup(2);creates the free group of rank 2. Here the generators may be referenced using the standard names, F.1 and F.2.
The statement
> F<x, y> := FreeGroup(2);defines F to be the free group of rank 2 and assigns the names x and y to the two generators.
Suppose G is an fp-group for which generators have already been defined. This subsection defines the elementary operations on words that are derived from the multiplication and inversion operators. The availability of the operators defined here enables the user to construct an element (word) of G in terms of the generators as follows:
Given words u and v belonging to the same fp-group G, return the product of u and v.
The n-th power of the word u, where n is an integer. When invoked with n = (-1), the function computes the inverse of u. When invoked with n = 0, the function returns the identity element.
Given words u and v belonging to the same fp-group G, return the conjugate v^(-1) * u * v of the word u by the word v.
Given words u and v belonging to the same fp-group G, return the commutator u^(-1)v^(-1)uv of the words u and v.
Given the n words u_1, ..., u_n belonging to the same fp-group G, return their commutator. Commutators are left-normed, so that they are evaluated from left to right.
Given a group G defined on r generators and a sequence [i_1, ..., i_s] of integers lying in the range [ - r, r], excluding 0, construct the word G.|i_1|^(epsilon_1) * G.|i_2|^(epsilon_2) * ... * G.|i_s|^(epsilon_s) where epsilon_j is +1 if i_j is positive, and -1 if i_j is negative.
Construct the identity element (empty word) for the fp-group G.
Given words w_1 and w_2 over the generators of an fp-group G, create the relation w_1 = w_2. Note that this relation is not automatically added to the existing set of defining relations R for G. It may be added to R, for example, through use of the quo-constructor (see below).
Given a relation r over the generators of G, return the left hand side of the relation r. The object returned is a word over the generators of G.
Given a relation r over the generators of G, return the right hand side of the relation r. The object returned is a word over the generators of G.
Redefine the left hand side of the relation r to be the word w.
Redefine the right hand side of the relation r to be the word w.
Given a homomorphism of the group G for which r is a relation, return the image of r under f.
Given a homomorphism of the group G for which r is a relation, construct the complete set of images of r under f.
Group over which the relation r is taken.
Simplify the word w as much as possible by replacing every occurrence of the LHS of r in w by its RHS.
Simplify the word w as much as possible by replacing every occurrence of the left hand side of some relation r, contained in the set or sequence of relations S, by its right hand side.
> F<x, y> := FreeGroup(2);
> rels := { x^2 = y^3, (x*y)^4 = Id(F) } ;
To replace one side of a relation, the easiest way is to reassign the relation. So for example, to replace the relation x^2=y^3 by x^2=y^4, we go:
> r := x^2 = y^3; > r := LHS(r) = y^4;
A group with non-trivial relations is constructed as a quotient of an
existing group, possibly a free group. For convenience, the necessary
free group may be constructed in-line.
quo< F | R > : GrpFP, List -> GrpFP, Hom(Grp)
Given an fp-group F, and a set of relations R in the generators of F, construct the quotient G of F by the normal subgroup of F defined by R. The group G is defined by means of a presentation which consists of the relations for F (if any), together with the additional relations defined by the list R.The expression defining F may be either simply the name of a previously constructed group, or an expression defining an fp-group.
If R is a list then each term of the list is either a word, a relation, a relation list or a subgroup of F.
A word is interpreted as a relator.
A relation consists of a pair of words, separated by `='. (See above.)
A relation list consists of a list of words, where each pair of adjacent words is separated by `=': w_1 = w_2 = ... = w_r. This is interpreted as the set of relations w_1 = w_r, ..., w_(r - 1) = w_r. Note that the relation list construct is only meaningful in the context of the quo-constructor.
A subgroup H appearing in the list R contributes its generators to the relation set for G, i.e. each generator of H is interpreted as a relator for G.
The group F may be referred to by the special symbol $ in any word appearing to the right of the `|' symbol in the quo-constructor. Also, in the context of the quo-constructor, the identity element (empty word) may be represented by the digit 1.
The function returns:
- The quotient group G;
- The natural homomorphism phi : F -> G.
Given a list X of variables x_1, ..., x_r, and a list of relations R over these generators, first construct the free group F on the generators x_1, ..., x_r and then construct the quotient of F corresponding to the normal subgroup of F defined by the relations R.The syntax for the relations R is the same as for the quo-constructor. The function returns:
- The quotient group G;
- The natural homomorphism phi : F -> G.
Given a subgroup H of the group G, construct the quotient of G by the normal closure N of H. The quotient is formed by taking the presentation for G and including the generating words of H as additional relators.
> F<a, b> := FreeGroup(2); > G<x, y>, phi := quo< F | a^2, b^3, (a*b)^4 >;Alternatively, giving the relations as a relations list, the presentation would be specified as:
> F<a, b> := FreeGroup(2); > G<x, y>, phi := quo< F | a^2 = b^3 = (a*b)^4 = 1>;Finally, giving the relations in the form of a set of relations, this presentation would be specified as:
> F<a, b> := FreeGroup(2);
> rels := { a^2, b^3, (a*b)^4 };
> G<x, y>, phi := quo< F | rels >;
> S4<x, y> := quo< FreeGroup(2) | $.1^2, $.2^3, ($.1*$.2)^4 >;
> S4;
Finitely presented group S4 on 2 generators
Relations
x^2 = Id(S4)
y^3 = Id(S4)
(x * y)^4 = Id(S4)
> G<r, s> := Group< r, s | r^3 = s^3 = (r*s)^2 >;
> G<r, s, t> := Group<r, s, t | r^2, s^2, t^2, r*s*t = s*t*r = t*r*s>;
> F<x, y> := Group< x, y | x^2 = y^3 = (x*y)^7 = 1 >;
> F;
Finitely presented group F on 2 generators
Relations
x^2 = Id(F)
y^3 = Id(F)
(x * y)^7 = Id(F)
> G<a, b> := quo< F | (x, y)^8 >;
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^3 = Id(G)
(a * b)^7 = Id(G)
(a, b)^8 = Id(G)
> Order(G);
10752
> Glmnk := func< l, m, n, k | Group< a, b | a^l, b^m, (a*b)^n, (a*b^-1)^k > >;
> G<a, b> := Glmnk(3, 3, 4, 4);
> G;
Finitely presented group G on 2 generators
Relations
a^3 = Id(G)
b^3 = Id(G)
(a * b)^4 = Id(G)
(a * b^-1)^4 = Id(G)
> Order(G);
168
> G<a, b> := Glmnk(2, 3, 4, 5);
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^3 = Id(G)
(a * b)^4 = Id(G)
(a * b^-1)^5 = Id(G)
> Order(G);
1
> // Thus (2, 3 | 4, 5 ) is the trivial group
Given an fp-group G with presentation < X | R >, create a new fp-group with presentation < X union { z } | R >, where z is a symbol not in X.
Given an fp-group G with presentation < X | R >, and given also a word w in the generators X, create a new fp-group having the presentation < X union { z } | R union { z = w } >, where z is a symbol not in X.
Given an fp-group G, and a relation r on the generators of G, create a new fp-group whose presentation consists of the relations of G together with the relation r.
Given an fp-group G, and an element g of G, create a new fp-group whose presentation consists of the relations of G together with the relation g=( Id(G)).
Given an fp-group G, and a relation r on the generators of G, create a new fp-group which has as its presentation the relations of G together with the relation r inserted after the i-th existing relation of G.
Given an fp-group G, and an element g of G, create a new fp-group which has as its presentation the relations of G together with the relation g=( Id(G)) inserted after the i-th existing relation of G.
Given an fp-group G with presentation < X | R >, and given also an element z in X, create a new fp-group with presentation < X - {z} | R' >, where the relations R' are obtained from R by deleting all relations containing an occurrence of z.
Given an fp-group G, which includes the relation r amongst its relations, create a new fp-group which has as its presentation the relations of G with relation r omitted.
Given an fp-group G, which includes the relation g=( Id(G)) amongst its relations, create a new fp-group which has as its presentation the relations of G with this relation omitted.
Given an fp-group G, create a new fp-group which has as its presentation the relations for G with the i-th relation deleted.
Given an fp-group G, which includes the relation s or h=( Id(G)) amongst its relations, create a new fp-group which has as its presentation the relations for G with the relation s replaced by the relation r or g=( Id(G)).
Given an fp-group G and a relation r in the generators of G, create a new fp-group which has as its presentation the relations for G with relation number i replaced by the relation r.
Given an fp-group G and an element g of G, create a new fp-group which has as its presentation the relations for G with relation number i replaced by the relation g=( Id(G)).
> G<x,y,z,h,k,a> := Group< x, y, z, h, k, a | > x^2, y^2, z^2, (x,y), (y,z), (x,z), h^3, k^3, (h,k), > (x,k), (y,k), (z,k), x^h*y, y^h*z, z^h*x, a^2, a*x*a*y, > a*y*a*x, (a,z), (a,k), (a*h)^2 >; > for i := 0 to 1 do > for j := 0 to 1 do > for k := 0 to 1 do > for l := 0 to 2 do > rel := G.1^i*G.2^j*G.3^k*G.5^l*(G.6*G.4)^2 = Id(G); > K := ReplaceRelation(G, 21, rel); > print Order(K), Index(K, sub< K | K.6, K.4>); > end for; > end for; > end for; > end for;<0, 0, 0, 0> 144 24 <0, 0, 0, 1> 144 8 <0, 0, 0, 2> 144 8 <0, 0, 1, 0> 18 3 <0, 0, 1, 1> 18 1 <0, 0, 1, 2> 18 1 <0, 1, 0, 0> 72 3 <0, 1, 0, 1> 72 1 <0, 1, 0, 2> 72 1 <0, 1, 1, 0> 36 6 <0, 1, 1, 1> 36 2 <0, 1, 1, 2> 36 2 <1, 0, 0, 0> 18 3 <1, 0, 0, 1> 18 1 <1, 0, 0, 2> 18 1 <1, 0, 1, 0> 144 6 <1, 0, 1, 1> 144 2 <1, 0, 1, 2> 144 2 <1, 1, 0, 0> 36 6 <1, 1, 0, 1> 36 2 <1, 1, 0, 2> 36 2 <1, 1, 1, 0> 72 12 <1, 1, 1, 1> 72 4 <1, 1, 1, 2> 72 4