route is taken, the car must travel at least |1 - (-2)| = 3 horizontal blocks and at least |2 - 4| = 2 vertical blocks, for a total trip distance of at least 3 + 2 = 5. If the car goes out of its way the trip may be longer, but the minimal distance travelled by the car, equal in value to the word metric between and is therefore equal to 5. In general, given two elements and of , the distance between and in the word metric is equal to . 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. Variations The word metric has an equivalent definition formulated in more geometric terms using the Cayley graph of G with respect to the generating set S. When each edge of the Cayley graph is assigned a metric of length 1, the distance between two group elements g,h in G is equal to the shortest length of a path in the Cayley graph from the vertex g to the vertex h. The word metric