Definition
Let G be a group, let S be a generating set for G, and suppose that S is closed under the inverse operation on G. A word over the set S is just a finite sequence whose entries are elements of S. The integer L is called the length of the word . Using the group operation in G, the entries of a word can be multiplied in order, remembering that the entries are elements of G. The result of this multiplication is an element in the group G which is called the evaluation of the word w. As a special case, the empty word has length zero, and its evaluation is the identity element of G.
Given an element g of G, its word norm |g| with respect to the generating set S is defined to be the shortest length of a word over S whose evaluation is equal to g. Given two elements g,h in G, the distance d(g,h) in the word metric with respect to S is defined to be . Equivalently, d(g,h) is the shortest length of a word w over S such that .
The word metric on G satisfies the axioms for a metric, and it is not hard to prove this. The proof of the symmetry axiom d(g,h) = d(h,g) for a metric uses the assumption that the generating set S is closed under inverse.
Read more about this topic: Word Metric
Famous quotes containing the word definition:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)