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group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a word is any written product of
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
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. Present ...
, and are central objects of study in
combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a nat ...
.


Definitions

Let ''G'' be a group, and let ''S'' be a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of ''G''. A word in ''S'' is any
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, o ...
of the form :s_1^ s_2^ \cdots s_n^ where ''s''1,...,''sn'' are elements of ''S'', called generators, 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 unique Uniqueness of identity element and inverses
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
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, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
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 line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
to denote inverses of elements of ''S''. Using overline notation, the above word would be written as follows: :x^2\overlinezyz^3\overline^2. \,


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 (defined
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
) that follow from the
group axioms In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Thes ...
. 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. A word is cyclically reduced
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
every
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
of the word is reduced.


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 reversing 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.


Generating set 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''. When ''S'' is not a generating set for ''G'', the set of elements represented by words in ''S'' is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of ''G'', known as the subgroup of ''G'' generated by ''S'' and usually denoted \langle S\rangle. It is the smallest subgroup of ''G'' that contains the elements of ''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 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 ...
with  and 1 representing the empty word (the identity element for the group). * 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 of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
Dih''n'' with  and 1 as above. * The set of words of the form ''xmyn'' for ''m,n'' ∈ Z are a normal form for the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s and with . * The set of reduced words in ''S'' are the unique normal form for the free group over ''S''.


Relations and presentations

If ''S'' is a generating set for a group ''G'', 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. 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.


Free groups

If ''S'' is any set, the free group 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''.


See also

*
Word problem (mathematics) In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other ...
*
Word problem for groups In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group ''G'' is the algorithmic problem of deciding whether two words in the generators represent the same elem ...


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