**Words in abstract generators** are a type of group elements in **GAP3**.
In the following we will abbreviate their full name to **abstract words**
or just to **words**.

A word is just a sequence of letters, where each letter is an abstract
generator or its inverse. Words are multiplied by concatenating them and
removing adjacent pairs of a generator and its inverse. Abstract
generators are created by the function `AbstractGenerator`

(see
AbstractGenerator).

Note that words do not belong to a certain group. Any two words can be
multiplied. In effect we compute with words in a free group of
potentially infinite rank (potentially infinite because we can always
create new abstract generators with `AbstractGenerator`

).

Words are entered as expressions in abstract generators and are displayed
as product of abstract generators (and powers thereof). The trivial word
can be entered and is displayed as `IdWord`

.

gap> a := AbstractGenerator( "a" ); a gap> b := AbstractGenerator( "b" ); b gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> a^0; IdWord

The first sections in this chapter describe the functions that create abstract generators (see AbstractGenerator and AbstractGenerators). The next sections define the operations for words (see Comparisons of Words and Operations for Words). The next section describes the function that tests whether an object is a word (see IsWord). The next sections describe the functions that compute the number of letters of a word (see LengthWord and ExponentSumWord). The next sections describe the functions that extract or find a subword (see Subword and PositionWord). The final sections describe the functions that modify words (see SubstitutedWord, EliminatedWord, and MappedWord).

Note that words in abstract generators are different from words in finite polycyclic groups (see Words in Finite Polycyclic Groups).

- AbstractGenerator
- AbstractGenerators
- Comparisons of Words
- Operations for Words
- IsWord
- LengthWord
- ExponentSumWord
- Subword
- PositionWord
- SubstitutedWord
- EliminatedWord
- MappedWord

`AbstractGenerator( `

`string` )

`AbstractGenerator`

returns a new abstract generator. This abstract
generator is printed using the string `string` passed as argument to
`AbstractGenerator`

.

gap> a := AbstractGenerator( "a" ); a gap> a^5; a^5

Note that the string is only used to print the abstract generator and to order abstract generators (see Comparisons of Words). It is possible for two different abstract generators to use the same string and still be different.

gap> b := AbstractGenerator( "a" ); a gap> a = b; false

Also when you define abstract generators interactively it is a good idea
to use the identifier of the variable as the name of the abstract
generator, because then what **GAP3** will output for a word is equal to
what you can input to obtain this word. The following is an example of
what you should probably **not** do.

gap> c := AbstractGenerator( "d" ); d gap> d := AbstractGenerator( "c" ); c gap> (c*d)^3; d*c*d*c*d*c gap> d*c*d*c*d*c; c*d*c*d*c*d

`AbstractGenerators( `

`string`, `n` )

`AbstractGenerators`

returns a list of `n` new abstract generators.
These new generators are printed using

, `string`1

, ...,
`string`2

.
`string`

gap> AbstractGenerators( "a", 3 ); [ a1, a2, a3 ]

`AbstractGenerators`

could be defined as follows (see
AbstractGenerator).

AbstractGenerators := function ( string, n ) local gens, i; gens := []; for i in [1..n] do Add( gens, AbstractGenerator( ConcatenationString( string, String(i) ) ) ); od; return gens; end;

`w1` = `w2`

`w1` <> `w2`

The equality operator `=`

evaluates to `true`

if the two words `w1` and
`w2` are equal and to `false`

otherwise. The inequality operator `<>`

evaluates to `true`

if the two words `w1` and `w2` are not equal and to
`false`

otherwise.

You can compare words with objects of other types, but they are never equal of course.

gap> a := AbstractGenerator( "a" );; gap> b := AbstractGenerator( "b" );; gap> a = b; false gap> (a^2*b)^5*b^-1 = a^2*b*a^2*b*a^2*b*a^2*b*a^2; true

`w1` < `w2`

`w1` <= `w2`

`w1` > `w2`

`w1` >= `w2`

The operators `<`

, `<=`

, `>`

, and `=>`

evaluate to `true`

if the word
`w1` is less than, less than or equal to, greater than, and greater than
or equal to the word `w2`.

Words are ordered as follows. One word `w1` is considered smaller than
another word `w2` it it is shorted, or, if they have the same length, if
it is first in the lexicographical ordering implied by the ordering of
the abstract generators. The ordering of abstract generators is as
follows. The abstract generators are ordered with respect to the strings
that were passed to `AbstractGenerator`

when creating these abstract
generators. Each abstract generator `g` is also smaller than its
inverse, but this inverse is smaller than any abstract generator that is
larger than `g`.

Words can also be compared with objects of other types. Integers, rationals, cyclotomics, finite field elements, and permutations are smaller than words, everything else is larger.

gap> IdWord<a; a<a^-1; a^-1<b; b<b^-1; b^-1<a^2; a^2<a*b; true true true true true true

`w1` * `w2`

The operator `*`

evaluates to the product of the two words `w1` and
`w2`. Note that words do not belong to a specific group, thus any two
words can be multiplied. Multiplication of words is done by
concatenating the words and removing adjacent pairs of an abstract
generator and its inverse.

`w1` / `w2`

The operator `/`

evaluates to the quotient *w1*w2 ^{-1}* of the two words

`w1` ^ `w2`

The operator `^`

evaluates to the conjugate *w2 ^{-1}* w1* w2* of the
word

`w1` ^ `i`

The powering operator `^`

returns the `i`-th power of the word `w1`,
where `i` must be an integer. If `i` is zero, the value is `IdWord`

.

`list` * `w1`

`w1` * `list`

In this form the operator `*`

returns a new list where each entry is the
product of `w1` and the corresponding entry of `list`. Of course
multiplication must be defined between `w1` and each entry of `list`.

`list` / `w1`

In this form the operator `/`

returns a new list where each entry is the
quotient of `w1` and the corresponding entry of `list`. Of course
division must be defined between `w1` and each entry of `list`.

`Comm( `

`w1`, `w2` )

`Comm`

returns the commutator *w1 ^{-1}* w2^{-1}* w1* w2* of two words

`LeftQuotient( `

`w1`, `w2` )

`LeftQuotient`

returns the left quotient *w1 ^{-1}* w2* of two words

`IsWord( `

`obj` )

`IsWord`

returns `true`

if the object `obj`, which may be an object of
arbitrary type, is a word and `false`

otherwise. Signals an error if
`obj` is an unbound variable.

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> IsWord( w ); true gap> a := (1,2,3);; gap> IsWord( a^2 ); false

`LengthWord( `

`w` )

`LengthWord`

returns the length of the word `w`, i.e., the number of
letters in the word.

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> LengthWord( w ); 14 gap> LengthWord( a^13 ); 13 gap> LengthWord( IdWord ); 0

`ExponentSumWord( `

`w`, `gen` )

`ExponentSumWord`

returns the number of times the generator `gen` appears
in the word `w` minus the number of times its inverse appears in `w`. If
`gen` and its inverse do no occur in `w`, 0 is returned. `gen` may also
be the inverse of a generator of course.

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> ExponentSumWord( w, a ); 10 gap> ExponentSumWord( w, b ); 4 gap> ExponentSumWord( (a*b*a^-1)^3, a ); 0 gap> ExponentSumWord( (a*b*a^-1)^3, b^-1 ); -3

`Subword( `

`w`, `from`, `to` )

`Subword`

returns the subword of the word `w` that begins at position
`from` and ends at position `to`. `from` and `to` must be positive
integers. Indexing is done with origin 1.

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> Subword( w, 5, 8 ); a*b*a^2

`PositionWord( `

`w`, `sub`, `from` )

`PositionWord`

returns the position of the first occurrence of the word
`sub` in the word `w` starting at position `from`. If there is no such
occurrence, `false`

is returned. `from` must be a positive integer.
Indexing is done with origin 1.

In other words, `PositionWord(`

returns the smallest
integer `w`,`sub`,`from`)`i` larger than or equal to `from` such that `Subword( `

(see Subword).
`w`, `i`,
`i`+LengthWord(`sub`)-1 ) = `sub`

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> PositionWord( w, a^2*b, 2 ); 4 gap> PositionWord( w, a*b^2, 2 ); false

`SubstitutedWord( `

`w`, `from`, `to`, `by` )

`SubstitutedWord`

returns a new word where the subword of the word `w`
that begins at position `from` and ends at position `to` is replaced by
the word `by`. `from` and `to` must be positive integers. Indexing is
done with origin 1.

In other words `SubstitutedWord(`

is the word
`w`,`from`,`to`,`by`)`Subword(`

(see Subword).
`w`,1,`from`-1) * `by` * Subword(`w`,`to`+1,LengthWord(`w`)

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> SubstitutedWord(w,5,8,b^-1); a^2*b*a^3*b*a^2

`EliminatedWord( `

`word`, `gen`, `by` )

`EliminatedWord`

returns a new word where each occurrence of the
generator `gen` is replaced by the word `by`.

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> EliminatedWord( w, b, b^2 ); a^2*b^2*a^2*b^2*a^2*b^2*a^2*b^2*a^2

`MappedWord( `

`w`, `gens`, `imgs` )

`MappedWord`

returns the new group element that is obtained by replacing
each occurrence of a generator `gen` in the list of generators `gens` by
the corresponding group element `img` in the list of group elements
`imgs`. The lists `gens` and `imgs` must of course have the same length.

gap> a := AbstractGenerator("a");; gap> b := AbstractGenerator("b");; gap> w := (a^2*b)^5*b^-1; a^2*b*a^2*b*a^2*b*a^2*b*a^2 gap> MappedWord( w, [a,b], [(1,2,3),(1,2)] ); (1,3,2)

If the images in `imgs` are all words, and some of them are equal to the
corresponding generators in `gens`, then those may be omitted.

gap> MappedWord( w, [a], [a^2] ); a^4*b*a^4*b*a^4*b*a^4*b*a^4

Note that the special case that the list `gens` and `imgs` have only
length 1 is handled more efficiently by `EliminatedWord`

(see
EliminatedWord).

gap3-jm

24 Apr 2021