Ingroup theory, aword is any written product ofgroup elements and their inverses. For example, ifx,y andz are elements of a groupG, thenxy,z⁻¹xzz andy⁻¹zxx⁻¹yz⁻¹ are words in the set {x, y, z}. Two different words may evaluate to the same value inG,^[1] or even in every group.^[2] Words play an important role in the theory offree groups andpresentations, and are central objects of study incombinatorial group theory.

LetG be a group, and letS be asubset ofG. Aword inS is anyexpression of the form

${\displaystyle s_1^{{\epsilon }_1}{s}_2^{{\epsilon }_2}\cdots s_n^{{\epsilon }_n}}$

wheres₁,...,s_n are elements ofS, calledgenerators, and eachε_i is ±1. The numbern is known as thelength of the word.

Each word inS represents an element ofG, namely the product of the expression. By convention, the unique^[3]identity element can be represented by theempty word, which is the unique word of length zero.

When writing words, it is common to useexponential notation as an abbreviation. For example, the word

${\displaystyle xx{y}^{-1}zyzzz{x}^{-1}{x}^{-1}}$

could be written as

${\displaystyle x^2{y}^{-1}zy{z}^3{x}^{-2}.}$

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 anoverline to denote inverses of elements ofS. Using overline notation, the above word would be written as follows:

${\displaystyle x^2\overline{y}zy{z}^3{\overline{x}}^2.}$

Any word in which a generator appears next to its own inverse (xx⁻¹ orx⁻¹x) can be simplified by omitting the redundant pair:

${\displaystyle y^{-1}zx{x}^{-1}y⟶y^{-1}zy.}$

This operation is known asreduction, and it does not change the group element represented by the word. Reductions can be thought of as relations (definedbelow) that follow from thegroup axioms.

Areduced word is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions:

${\displaystyle xz{y}^{-1}x{x}^{-1}y{z}^{-1}z{z}^{-1}yz⟶xyz.}$

The result does not depend on the order in which the reductions are performed.

A word iscyclically reducedif and only if everycyclic permutation of the word is reduced.

Theproduct of two words is obtained by concatenation:

${\displaystyle \left(xzy{z}^{-1}\right)\left(z{y}^{-1}{x}^{-1}y\right)=xzy{z}^{-1}z{y}^{-1}{x}^{-1}y.}$

Even if the two words are reduced, the product may not be.

Theinverse of a word is obtained by inverting each generator, and reversing the order of the elements:

${\displaystyle {\left(z{y}^{-1}{x}^{-1}y\right)}^{-1}=y^{-1}xy{z}^{-1}.}$

The product of a word with its inverse can be reduced to the empty word:

${\displaystyle z{y}^{-1}{x}^{-1}y{y}^{-1}xy{z}^{-1}=1.}$

You can move a generator from the beginning to the end of a word byconjugation:

${\displaystyle x^{-1}\left(x{y}^{-1}{z}^{-1}yz\right)x=y^{-1}{z}^{-1}yzx.}$

A subsetS of a groupG is called agenerating set if every element ofG can be represented by a word inS.

WhenS is not a generating set forG, the set of elements represented by words inS is asubgroup ofG, known as thesubgroup ofG generated byS and usually denoted${\displaystyle ⟨S⟩}$. It is the smallest subgroup ofG that contains the elements ofS.

Anormal form for a groupG with generating setS is a choice of one reduced word inS for each element ofG. For example:

• The words 1,i,j,ij are a normal form for theKlein four-group withS = {i,  j}  and 1 representing the empty word (the identity element for the group).
• The words 1,r,r², ...,r^n-1,s,sr, ...,sr^n-1 are a normal form for thedihedral group Dih_n withS = {s,  r}  and 1 as above.
• The set of words of the formx^myⁿ form,n ∈ Z are a normal form for thedirect product of thecyclic groups⟨x⟩ and⟨y⟩ withS = {x,  y}.
• The set of reduced words inS are the uniquenormal form for the free group overS.

IfS is a generating set for a groupG, arelation is a pair of words inS that represent the same element ofG. These are usually written as equations, e.g.${\displaystyle x^{-1}yx=y^2.}$A set${\displaystyle \mathcal{R}}$ of relationsdefinesG if every relation inG follows logically from those in${\displaystyle \mathcal{R}}$ using theaxioms for a group. Apresentation forG is a pair${\displaystyle ⟨S\mid \mathcal{R}⟩}$, whereS is a generating set forG and${\displaystyle \mathcal{R}}$ is a defining set of relations.

For example, theKlein four-group can be defined by the presentation

${\displaystyle ⟨i,j\mid i^2=1,j^2=1,ij=ji⟩.}$

Here 1 denotes the empty word, which represents the identity element.

IfS is any set, thefree group overS is the group with presentation${\displaystyle ⟨S\mid ⟩}$. That is, the free group overS is the group generated by the elements ofS, with no extra relations. Every element of the free group can be written uniquely as a reduced word inS.

• Word problem (mathematics)
• Word problem for groups

• Epstein, David;Cannon, J. W.; Holt, D. F.; Levy, S. V. F.;Paterson, M. S.;Thurston, W. P. (1992).Word Processing in Groups. AK Peters.ISBN 0-86720-244-0..
• Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory".Trudy Mat. Inst. Steklov (in Russian).44:1–143.
• Robinson, Derek John Scott (1996).A course in the theory of groups. Berlin: Springer-Verlag.ISBN 0-387-94461-3.
• Rotman, Joseph J. (1995).An introduction to the theory of groups. Berlin: Springer-Verlag.ISBN 0-387-94285-8.
• Schupp, Paul E;Lyndon, Roger C. (2001).Combinatorial group theory. Berlin: Springer.ISBN 3-540-41158-5.
• Solitar, Donald;Magnus, Wilhelm; Karrass, Abraham (2004).Combinatorial group theory: presentations of groups in terms of generators and relations. New York: Dover.ISBN 0-486-43830-9.
• Stillwell, John (1993).Classical topology and combinatorial group theory. Berlin: Springer-Verlag.ISBN 0-387-97970-0.

• Combinatorial group theory
• Group theory
• Combinatorics on words

• CS1 Russian-language sources (ru)