Word (group Theory) - Reduced Words

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:

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 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:

The result does not depend on the order in which the reductions are performed.

If S is any set, the free group over S is the group with presentation . That is, the free group over S is the group generated by the elements of S, with no extra relations. Every element of the free group can be written uniquely as a reduced word in S.

A word is cyclically reduced if and only if every cyclic permutation of the word is reduced.