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,^[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.

^ for example, f_dr₁ and r₁f_c in the group of square symmetries
^ for example, xy and xzz⁻¹y

[1] r example, f_dr₁ and r₁f_c in the group of square symmetries

[2] r example, xy and xzz⁻¹y

[1]

[2]