Algebra

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Math 150a: Modern Algebra Homework 1 This problem set would ideally have been due Wednesday, October 3. But since I am running late, it can be turned with no penalty on Friday, October 5. Do problems 2.1.4 and 2.1.6 (see also the note in GK4) in addition to the following: GK1. Finish the multiplication table for the symmetries of a triangle that I started in class. GK2. Describe the symmetries of a square, just as I did an equilateral triangle. (You don’t have to write out the multiplication table though.) GK3. In general in computations in a group, you encounter words in elements and their inverses, such as the conjugate aba - 1 and the commutator aba - 1 b - 1 . Such a word is in reduced form if a letter never appears next to its inverse (because such a word would automatically simplify). For example,
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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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