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PROBLEM SET Homework Problems Math 3360, Fall 2013 November 20, 2013 • Write all of your answers on separate sheets of paper. You can keep the question sheet. • You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not √ approximations (e.g., 2, not 1.414). • This problem et has 5 problems. There are 240 points total. Good luck! Problem 1. In this problem, we’ll be working in Z12 . I’ll denote the elements ¯ of Z12 with bars: 0̄, 1̄, 2̄, . . . , 11. A. List all the elements of the cyclic subgroup H of Z12 generated by 3̄. B. List the elements of all of the cosets α + H, where α ∈ Z12 . How many distinct cosets are there? C. List all of the sums x + y for x ∈ (6̄ + H) and y ∈ (8̄ + H). Form the set of all these sums. Which coset is it? Problem 2. In this problem, we are working with the symmetric groups S4 and S5 . You might want to practice with S3 . A. List all of the possible cycle type for permutations in S4 . Use this information to list all the elements of each of the conjugacy classes in S4 . For each class, determine if the permutations in the class are all odd or all even. B. There are too many elements of S5 to list them all, but determine the possible cycle types, the number of conjugacy classes, the number of elements in each class and the parity (even or odd) of the elements of each class. Problem 3. In each part, we want to construct a necklace using 2 blue beads, with the rest of the beads being red. Of course, two patterns are considered equivalent if we can move one to the other using the action of the appropriate dihedral group. (This is similar to Example 4.3.4.) Determine the number of distinct patterns and give an example of each of them. See my website for figures. A. A necklace with 7 beads. B. A necklace with 8 beads. Problem 4. How many ways are there of coloring a cube so that two faces are blue, two faces are red and the remaining two faces are green? Of course, two colorings are considered equivalent if we can get one from the other by a rigid motion of the cube. Give an example of each distinct pattern. 1 Problem 5. Let G be a group. Recall that Z(G), the center of G, is the subgroup consisting of all elements that commute with everything in G. Show that if G/Z(G) is cyclic, then G is abelian. Remarks: Of course, if G is abelian, Z(G) = G, so G/Z(G) = {e}. Thus, we have shown that G/Z(G) can not be a nontrivial cyclic group. In particular, the order of G/Z(G) can’t be a prime. 2