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}.
[2] 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,...,sn are elements of S, called generators, and each εi is ±1.
When writing words, it is common to use exponential notation as an abbreviation.
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: 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: 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.
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 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.
For example, the Klein four-group can be defined by the presentation Here 1 denotes the empty word, which represents the identity element.
Every element of the free group can be written uniquely as a reduced word in S.