WORD (GROUP THEORY)

In group theory, a 'word' is any written product of group elements and their inverses. For example, if ''x'', ''y'', and ''z'' are elements of a group ''G'', then ''xy'', ''z''^-1''xzz'', and ''y''^-1''zxx''^-1''yz''^-1 are words in the set {''x'', ''y'', ''z''}. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory.

Contents

Definition

Notation

Words and presentations

Reduced words

Normal forms

Operations on words

The word problem

See also

Notes

References

Definition

Let ''G'' be a group, and let ''S'' be a subset of ''G''. A 'word in ''S''' is any expression of the form
:$s\_1^\{epsilon\_1\}\; s\_2^\{epsilon\_2\}\; cdots\; s\_n^\{epsilon\_n\}$
where ''s''₁,...,''s_n'' are elements of ''S'' and each ''ε_i'' is ±1. The number ''n'' is known as the 'length' of the word.
Each word in ''S'' represents an element of ''G'', namely the product of the expression. By convention, the identity element can be represented by the 'empty word', which is the unique word of length zero.

Notation

When writing words, it is common to use exponential notation as an abbreviation. For example, the word
:$x\; x\; y^\{-1\}\; z\; y\; z\; z\; z\; x^\{-1\}\; x^\{-1\}\; ,$
could be written as
:$x^2\; y^\{-1\}\; z\; y\; z^3\; x^\{-2\}\; ,$.
This latter expression is not a word itself—it is simply a shorter notation for the original.
When dealing with long words, it can be helpful to use an overline to denote inverses of elements of ''S''. Using overline notation, the above word would be written as follows:
:$x^2overline\{y\}zyz^3overline\{x\}^2\; ,$.

Words and presentations

Main articles: Presentation of a group

A subset ''S'' of a group ''G'' is called a generating set if every element of ''G'' can be represented by a word in ''S''. If ''S'' is a generating set, a 'relation' is a pair of words in ''S'' that represent the same element of ''G''. These are usually written as equations:
:$x^\{-1\}\; y\; x\; =\; y^2,$
A set $mathcal\{R\}$ of relations 'defines ''G''' if every relation in ''G'' follows logically from those in $mathcal\{R\}$, using the axioms for a group. A 'presentation' for ''G'' is a pair $langle\; S\; mid\; mathcal\{R\}$
angle, where ''S'' is a generating set for ''G'' and $mathcal\{R\}$ is a defining set of relations.
For example, the Klein four-group can be defined by the presentation
:$langle\; i,j\; mid\; i^2\; =\; 1,,j^2\; =\; 1,,ij=ji$
angle
Here 1 denotes the empty word, which represents the identity element.
When ''S'' is not a generating set for ''G'', the set of elements represented by words in ''S'' is a subgroup of ''G''. This is known as the 'subgroup of ''G'' generated by ''S''', and is usually denoted $langle\; S$
angle. It is the smallest subgroup of ''G'' that contains the elements of ''S''.

Reduced words

Any word in which a generator appears next to its own inverse (''xx''^-1 or ''x''^-1''x'') can be simplified by omitting the redundant pair:
:$y^\{-1\}zxx^\{-1\}y;;longrightarrow;;y^\{-1\}zy$
This operation is known as 'reduction', and it does not change the group element represented by the word. (Reductions can be thought of as relations that follow from the group axioms.)
A 'reduced word' is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions:
:$xzy^\{-1\}xx^\{-1\}yz^\{-1\}zz^\{-1\}yz;;longrightarrow;;xyz$
The result does not depend on the order in which the reductions are performed.
If ''S'' is any set, the free group over ''S'' is the group with presentation $langle\; Smid;$
angle. That is, the free group over ''S'' is the group generated by the elements of ''S'', with no extra relations. Every element of the free group can be written uniquely as a reduced word in ''S''.

Normal forms

A 'normal form' for a group ''G'' with generating set ''S'' is a choice of one reduced word in ''S'' for each element of ''G''. For example:

★ The words 1, ''i'', ''j'', ''ij'' are a normal form for the Klein four-group.

★ The words 1, ''r'', ''r''², ..., ''r^n-1'', ''s'', ''sr'', ''sr^n-1'' are a normal form for the dihedral group Dih_''n''.

★ The set of reduced words in ''S'' are a normal form for the free group over ''S''.

★ The set of words of the form ''x^myⁿ'' for ''m,n'' ∈ 'Z' are a normal form for the direct product of the cyclic groups ⟨''x''⟩ and ⟨''y''⟩.

Operations on words

The 'product' of two words is obtained by concatenation:
:$left(xzyz^\{-1\}$
ight)left(zy^{-1}x^{-1}y
ight) = xzyz^{-1}zy^{-1}x^{-1}y
Even if the two words are reduced, the product may not be.
The 'inverse' of a word is obtained by inverting each generator, and switching the order of the elements:
:$left(zy^\{-1\}x^\{-1\}y$
ight)^{-1}=y^{-1}xyz^{-1}
The product of a word with its inverse can be reduced to the empty word:
:$zy^\{-1\}x^\{-1\}y\; ;\; y^\{-1\}xyz^\{-1\}\; ;\; longrightarrow;;1$
You can move a generator from the beginning to the end of a word by conjugation:
:$x^\{-1\}left(xy^\{-1\}z^\{-1\}yz$
ight)x = y^{-1}z^{-1}yzx

The word problem

Main articles: Word problem for groups

Given a presentation $langle\; Smid\; mathcal\{R\}$
angle for a group ''G'', the 'word problem' is the algorithmic problem of deciding, given as input two words in ''S'', whether they represent the same element of ''G''. The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911. It was shown by Pyotr Sergeyevich Novikov in 1955 that there exists a finitely presented group ''G'' such that the word problem for ''G'' is undecidable.^[1]

See also

★ Group (mathematics)

★ Presentation of a group

★ Free group

Notes

1. P. S. Novikov. "On the algorithmic unsolvability of the word problem in group theory." ''Trudy Mat. Inst. Steklov'' '44' (1955), pp. 1-143. (in Russian)

References

★ .

★ Combinatorial group theory: presentations of groups in terms of generators and relations, Solitar, Donald; Magnus, Wilhelm; Karrass, Abraham, , , Dover Publications, 2004,

★ Combinatorial group theory, Schupp, Paul E.; Lyndon, Roger C., , , Springer, 2001,

★ Classical topology and combinatorial group theory, Stillwell, John, , , Springer-Verlag, 1993,

★ A course in the theory of groups, Robinson, Derek John Scott, , , Springer-Verlag, 1996,

★ An introduction to the theory of groups, Rotman, Joseph J., , , Springer-Verlag, 1995,