# 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*}. Two different words may evaluate to the same value in *G*,[1] or even in every group.[2] Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory.

## Definition

Let *G* be a group, and let *S* be a subset of *G*. A **word in S** is any expression of the form

where *s*_{1},...,*s _{n}* are elements of

*S*and each

*ε*is ±1. The number

_{i}*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)[3] 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

could be written as

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:

## Words and presentations

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, e.g.
A set of relations **defines G** if every relation in

*G*follows logically from those in , using the axioms for a group. A

**presentation**for

*G*is a pair , where

*S*is a generating set for

*G*and is a defining set of relations.

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

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 . 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:

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:

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 . 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 every cyclic permutation 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. - The words 1,
*r*,*r*^{2}, ...,*r*,^{n-1}*s*,*sr*, ...,*sr*are a normal form for the dihedral group Dih^{n-1}_{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*for^{m}y^{n}*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:

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:

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

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

## The word problem

Given a presentation 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 Novikov in 1955 that there exists a finitely presented group *G* such that the word problem for *G* is undecidable.(Novikov 1955)

## References

- 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.

- for example, f
_{d}r_{1}and r_{1}f_{c}in the group of square symmetries - for example,
*xy*and*xzz*^{−1}*y* - Uniqueness of identity element and inverses