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. 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,...,sn are elements of S 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 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