Word (group theory)
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In
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, a word is any written product of
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
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 . Two different words may evaluate to the same value in ''G'', or even in every group. Words play an important role in the theory of
free group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and
presentations A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
, and are central objects of study in
combinatorial group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.


Definition

Let ''G'' be a group, and let ''S'' be a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
of ''G''. A word in ''S'' is any expression of the form :s_1^ s_2^ \cdots s_n^ where ''s''1,...,''sn'' are elements of ''S'' and each ''εi'' 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 (unique) Uniqueness of identity element and inverses 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 Exponential may refer to any of several mathematical topics related to exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and ...
notation as an abbreviation. For example, the word :x x y^ z y z z z x^ x^ \, could be written as :x^2 y^ z y z^3 x^. \, 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 An overline, overscore, or overbar, is a typographical feature of a horizontal and vertical, horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a ''vinculum (symbol), vinculum'', a notation f ...

overline
to denote inverses of elements of ''S''. Using overline notation, the above word would be written as follows: :x^2\overlinezyz^3\overline^2. \,


Words and presentations

A subset ''S'' of a group ''G'' is called a
generating set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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, e.g. x^ y x = y^2.\, A set \mathcal of relations defines ''G'' if every relation in ''G'' follows logically from those in \mathcal, using the
axioms for a group
axioms for a group
. A presentation for ''G'' is a pair \langle S \mid \mathcal\rangle, where ''S'' is a generating set for ''G'' and \mathcal is a defining set of relations. For example, the
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
can be defined by the presentation :\langle i,j \mid i^2 = 1,\,j^2 = 1,\,ij=ji\rangle. 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 In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
of ''G''. This is known as the subgroup of ''G'' generated by ''S'', and is usually denoted \langle S\rangle. 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^zxx^y\;\;\longrightarrow\;\;y^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^xx^yz^zz^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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
over ''S'' is the group with presentation \langle S\mid\;\rangle. 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''. A word is cyclically reduced
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
every
cyclic permutation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the word is reduced.


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 In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
. * The words 1, ''r'', ''r''2, ..., ''rn-1'', ''s'', ''sr'', ..., ''srn-1'' are a normal form for the
dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
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 ''xmyn'' for ''m,n'' ∈ Z are a normal form for the
direct product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
s 〈''x''〉 and 〈''y''〉.


Operations on words

The product of two words is obtained by concatenation: :\left(xzyz^\right)\left(zy^x^y\right) = xzyz^zy^x^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^x^y\right)^=y^xyz^. The product of a word with its inverse can be reduced to the empty word: :zy^x^y \; y^xyz^ = 1. You can move a generator from the beginning to the end of a word by
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
: :x^\left(xy^z^yz\right)x = y^z^yzx.


The word problem

Given a presentation \langle S\mid \mathcal \rangle 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 Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, se ...

Max Dehn
in 1911. It was shown by
Pyotr Novikov Pyotr Sergeyevich Novikov (russian: Пётр Серге́евич Но́виков; 15 August 1901, Moscow Moscow ( , American English, US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐˈskva, a=Москва.ogg) is the Capital city, ...
in 1955 that there exists a finitely presented group ''G'' such that the word problem for ''G'' is undecidable.


Notes


References

*. * * * * * *{{cite book , author=Stillwell, John , title=Classical topology and combinatorial group theory , publisher=Springer-Verlag , location=Berlin , year=1993, isbn=0-387-97970-0 Combinatorial group theory Group theory Combinatorics on words