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−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. 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 groups and presentations, and are central objects of study in combinatorial group theory.
Definitions 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 ε 1 s 2 ε 2 ⋯ s n ε n {\displaystyle s_{1}^{\varepsilon _{1}}s_{2}^{\varepsilon _{2}}\cdots s_{n}^{\varepsilon _{n}}}
where s1,...,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 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 {\displaystyle xxy^{-1}zyzzzx^{-1}x^{-1}\,}
could be written as
x 2 y − 1 z y z 3 x − 2 . {\displaystyle x^{2}y^{-1}zyz^{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 2 y ¯ z y z 3 x ¯ 2 . {\displaystyle x^{2}{\overline {y}}zyz^{3}{\overline {x}}^{2}.\,}
Reduced words Any word in which a generator appears next to its own inverse (xx−1 or x−1x) can be simplified by omitting the redundant pair:
y − 1 z x x − 1 y ⟶ y − 1 z y . {\displaystyle 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 (defined below) 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:
x z y − 1 x x − 1 y z − 1 z z − 1 y z ⟶ x y z . {\displaystyle 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. A word is cyclically reduced if and only if every cyclic permutation of the word is reduced.
Operations on words The product of two words is obtained by concatenation:
( x z y z − 1 ) ( z y − 1 x − 1 y ) = x z y z − 1 z y − 1 x − 1 y . {\displaystyle \left(xzyz^{-1}\right)\left(zy^{-1}x^{-1}y\right)=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 reversing the order of the elements:
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