**Unformatted text preview: **ba-1 ab is not in reduced form. How many words of length n are there in a and b and their inverses that are in reduced form? GK4. As I read it, Problem 2.1.6 asks for all full parenthesizations of the word abcd that reduce this product to multiplying pairs of elements. For example, (( ab ) c ) d is one of them; there are 5 in total. Now that you know all 5, connect every pair of them by an edge when they differ by a single application of the associative law. (The essential proposition 1.4 says that the graph that you get this way is connected, for words of any length.) GK5. (Extra credit) Do problem GK4 for parenthesizations of abcde . Amazingly enough, the result is a planar graph. ( I.e. , it can be drawn without making any of the edges cross.)...

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