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Free Groups and Words

Subsections

Construction of a Free Group

FreeGroup(n) : RngIntElt -> GrpFP
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< v1, ..., vn > := FreeGroup(n);


Example GrpFP_1_Free (H70E1)

The statement

> 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.

Construction of Words

The operations in this section apply to both, free groups and arbitrary fp-groups.

G ! [ i1, ..., is ] : GrpFP, [ RngIntElt ] -> GrpFPElt
Given a group G defined on r generators and a sequence [i1, ..., is] of integers lying in the range [ - r, r], excluding 0, construct the word G.|i1|ε1 * G.|i2|ε2 * ... * G.|is|εs where εj is +1 if ij is positive, and -1 if ij is negative.
Identity(G) : GrpFP -> GrpFPElt
Id(G) : GrpFP -> GrpFPElt
G ! 1 : GrpFP, RngIntElt -> GrpFPElt
Construct the identity element, represented as the empty word, for the fp-group G. For a sample application of this function, see Example H70E3.
Random(G, m, n) : GrpFP, RngIntElt, RngIntElt -> GrpFPElt
A random word of length l in the generators of the group G, where m ≤l ≤n. For a sample application of this function, see Example H70E3.

Access Functions for Words

This section describes some basic access functions for words. These operations apply to both, free groups and arbitrary fp-groups.

# w : GrpFPElt -> RngIntElt
The length of the word w.
ElementToSequence(w) : GrpFPElt -> [ RngIntElt ]
Eltseq(w) : GrpFPElt -> [ RngIntElt ]
The sequence Q obtained by decomposing the word w into its constituent generators and generator inverses. Suppose w is a word in the group G. Then, if w = G.i1e1 ... G.imem, with each ei equalling plus or minus 1, then Q[j] = ij if ej = + 1 and Q[j] = - ij if ej = (-1), for j = 1, ..., m.
ExponentSum(w, x) : GrpFPElt, GrpFPElt -> RngIntElt
Weight(w, x) : GrpFPElt, GrpFPElt -> RngIntElt
Given a word w, and the name of a generator x of a group G, compute the sum of the exponents of the generator x in the word w. For a sample application of this function, see Example H70E3.
GeneratorNumber(w) : GrpFPElt -> RngIntElt
Suppose w is a word belonging to a group G. Assume x is the name of the i-th generator of G. Then
(i)
if w = Identity(G), GeneratorNumber(w) is 0;
(ii)
if w = x * w', w' a word in G, GeneratorNumber(w) is i;
(iii)
if w = x - 1 * w', w' a word in G, GeneratorNumber(w) is -i.
LeadingGenerator(w) : GrpFPElt -> GrpFPElt
Suppose w is a word belonging to a group G. If w = xε * w', w' a word in G, x a generator of G and ε∈{-1, + 1}, the functions returns xε. If w= Identity(G), it returns Identity(G). For a sample application of this function, see Example H70E3.
Parent(w) : GrpFPElt -> GrpFP
The parent group G of the word w.

Example GrpFP_1_WordAccess (H70E2)

Consider the free group F on 6 generators

> F<u,v,w,x,y,z> := FreeGroup(6);
and the sequence of words

> rels := [ (u*v)^42, (v,x), (x*z^2)^4,
>           v^2*y^3, (v*z)^3, y^4, (x*z)^3 ];
The abelianised relation matrix of the quotient < u, v, w, x, y, z | (uv)42, (v, x), (xz2)4, v2y3, (vz)3, y4, (xz)3 > can be obtained using the following construction

> R := Matrix(Integers(), 
>        [ [ Weight(r, F.j) : j in [1..6] ] : r in rels ]);
> R;
[42 42  0  0  0  0]
[ 0  0  0  0  0  0]
[ 0  0  0  4  0  8]
[ 0  2  0  0  3  0]
[ 0  3  0  0  0  3]
[ 0  0  0  0  4  0]
[ 0  0  0  3  0  3]
(The function Matrix constructs a matrix from a sequence of row vectors.)

Arithmetic Operators for Words

Suppose G is an fp-group for which generators have already been defined. This subsection defines the elementary arithmetic 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:

(i)
A generator is a word;
(ii)
The expression (u) is a word, where u is a word;
(iii)
The product u * v of the words u and v is a word;
(iv)
The conjugate uv of the word u by the word v, is a word (uv expands into the word v - 1 * u * v);
(v)
The power of a word, un, where u is a word and n is an integer, is a word;
(vi)
The commutator (u, v) of the words u and v is a word ((u, v) expands into the word u - 1 * v - 1 * u * v). Note that (u, v, w) is equivalent to ((u, v), w), i.e. commutators are left-normed.

The word operations defined here may be applied either to the words of a free group or the words of a group with non-trivial relations. If such an operator is applied to a group possessing non-trivial relations, only free reduction will be applied to the resulting words.
u * v : GrpFPElt, GrpFPElt -> GrpFPElt
Given words u and v belonging to the same fp-group G, return the product of u and v.
u ^ n : GrpFPElt, RngIntElt -> GrpFPElt
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.
u ^ v : GrpFPElt, GrpFPElt -> GrpFPElt
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.
(u, v) : GrpFPElt, GrpFPElt -> GrpFPElt
Given words u and v belonging to the same fp-group G, return the commutator u - 1v - 1uv of the words u and v.
(u1, ..., un) : List(GrpFPElt) -> GrpFPElt
Given the n words u1, ..., un belonging to the same fp-group G, return their commutator. Commutators are left-normed, so that they are evaluated from left to right.

Comparison of Words

Words in an fp-group may be compared both for equality and for their relationship with respect to a natural lexicographic ordering. It should be noted that even when a pair of words belong to a group defined by non-trivial relations, only the free reductions of the words are compared. Thus, a pair of words belonging to a group G may be declared to be distinct even though they may represent the same element of G.

The words of an fp-group G are ordered first by length and then lexicographically. The lexicographic ordering is determined by the following ordering on the generators and their inverses: G.1 < G.1 - 1 < G.2 < G.2 - 1 < ... Here, u and v are words belonging to some common fp-group.

u eq v : GrpFPElt, GrpFPElt -> BoolElt
Return true if the free reductions of the words u and v are identical.
u ne v : GrpFPElt, GrpFPElt -> BoolElt
Return true if the free reductions of the words u and v are not identical.
u lt v : GrpFPElt, GrpFPElt -> BoolElt
Return true if the word u precedes the word v, with respect to the ordering defined above for elements of an fp-group. The words u and v are freely reduced before the comparison is made.
u le v : GrpFPElt, GrpFPElt -> BoolElt
Return true if the word u either precedes, or is equal to, the word v, with respect to the ordering defined above for elements of an fp-group. The words u and v are freely reduced before the comparison is made.
u ge v : GrpFPElt, GrpFPElt -> BoolElt
Return true if the word u either follows, or is equal to, the word v, with respect to the ordering defined above for elements of an fp-group. The words u and v are freely reduced before the comparison is made.
u gt v : GrpFPElt, GrpFPElt -> BoolElt
Return true if the word u follows the word v, with respect to the ordering defined above for elements of an fp-group. The words u and v are freely reduced before the comparison is made.

Example GrpFP_1_Words (H70E3)

We construct the free group on three generators and generate a random word

w of length between 4 and 6.

> F<a,b,c> := FreeGroup(3);
> 
> w := Random(F, 4, 6);
> w;
b^-1 * a^-1 * b^2 * a^-1
We print the length of w and the weight (the exponent sum) of generator a in w.

> #w;
5
> 
> Weight(w, a);
-2
We now strip the generators from w one by one, using arithmetic and comparison operators for words and the access function LeadingGenerator.

> while w ne Identity(F) do
>    g := LeadingGenerator(w);
>    print g;
>    w := g^-1 * w;
> end while;
b^-1
a^-1
b
b
a^-1

Relations

A relation is an equality between two words in a fp-group. To facilitate working with relations, a relation type is provided.

w1 = w2 : GrpFPElt, GrpFPElt -> GrpFPRel
Given words w1 and w2 over the generators of an fp-group G, create the relation w1 = w2. 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).
r[1] : RelElt, RngIntElt -> GrpFPElt
LHS(r) : RelElt -> GrpFPElt
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.
r[2] : RelElt, RngIntElt -> GrpFPElt
RHS(r) : RelElt -> GrpFPElt
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.
r[1] = w : GrpFPRel, RngIntElt, GrpFPElt -> GrpFPRel
Redefine the left hand side of the relation r to be the word w.
r[2] = w : GrpFPRel, RngIntElt, GrpFPElt -> GrpFPRel
Redefine the right hand side of the relation r to be the word w.
f(r) : Hom(GrpFP), GrpFPRel -> GrpFPRel
Given a homomorphism of the group G for which r is a relation, return the image of r under f.
Parent(r) : RelElt -> GrpFP
Group over which the relation r is taken.

Example GrpFP_1_Relations (H70E4)

We may define a group and a set of relations as follows:

> 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 x2=y3 by x2=y4, we go:

> r := x^2 = y^3;
> r := LHS(r) = y^4;

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013