1. Problem Set 11.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.1. Problem Set 1The goal is to try all the problems and do half of them.1.1. Suppose that σ is a permutation on n with k cycles. Then showthat the composition (12)σ has k, k− 1 or k+1 cycles. Give an exampleof each.1.2. Suppose that a, b are permutations and x, y, z are also permuta-tions (or: take a, b, x, y, z to be elements of any group G). Supposethatxab = ayb = abz.(1) Show that x, y, z are conjugate to each other.(2) When are x, y, z all different? Give an example when they aredifferent and when two of them are the same.1.3. The permutation (13) can be written as a product of simple trans-position (i, i + 1) in two different ways. Draw a picture and find thesetwo expressions. Can you prove that there are no other expressions for(13) with the same length?1.4. Same question for (14).1.5. Suppose that α, β are two vectors in R3and θ is the angle betweenthem. Let rα, rβbe the reflections along α, β respectively.(1) Describe the compositions rα◦ rβand rβ◦ rαgeometrically.(They are rotations. What is the axis of rotation and whatangle do they rotate?) Draw a picture.(2) Conclude that, for a root system in R2, the lengths of the rootsalternate between two lengths as you go around a circle.(3) When do rα, rβcommute?1.6. The order of a permutation σ is the smallest positive integer mso that σm= e. For example, the order of the elementσ = (12)(345)is 6. The formula is: the order of σ is the least common multiple ofthe lengths of the cycles in σ. Why is this true?121.7. Find an interpretation for the areas under a binary tree. Do allpossible areas occur? Do the areas determine the tree?(The way it is drawn here, the numbers are the areas plus12.)@@@u@@@@@@@@@u1 4 1 3 1