Group Theory
Walther Franz Anton von Dyck began the work of combinatorics on words in group theory by his published work in 1882 and 1883. He began by using words as group elements. Lagrange also contributed in 1771 with his work on permutation groups.
One aspect of combinatorics on words studied in group theory is reduced words. A group is constructed with words on some alphabet including generators and inverse elements, excluding factors that appear of the form aā or āa, for some a in the alphabet. Reduced words are formed when the factors aā, āa are used to cancel out elements until a unique word is reached.
Nielsen transformations were also developed. For a set of elements of a free group, a Nielsen transformation is achieved by three transformations; replacing an element with its inverse, replacing an element with the product of itself and another element, and eliminating any element equal to 1. By applying these transformations Nielsen reduced sets are formed. A reduced set means no element can be multiplied by other elements to cancel out completely. There are also connections with Nielsen transformations with Sturmian words.
Read more about this topic: Combinatorics On Words
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