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⁻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.

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

where s₁,...,s_n are elements of S and each ε_i 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) element can be represented by the empty word, which is the unique word of length zero.

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:

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. ${\displaystyle x^{-1}yx=y^{2}.\,}$ A set ${\displaystyle {\mathcal {R}}}$ of relations defines G if every relation in G follows logically from those in ${\displaystyle {\mathcal {R}}}$, using the axioms for a group. A presentation for G is a pair ${\displaystyle \langle S\mid {\mathcal {R}}\rangle }$, where S is a generating set for G and ${\displaystyle {\mathcal {R}}}$ is a defining set of relations.

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