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Math 366 Assignment 3 Due Wednesday, November 3 1. What is the order 1 2 3 (a) 2 1 5 1 2 3 (b) 7 6 1 1 2 3 (c) 1 2 3 1 2 2. Let α = 2 1 of each of the following permutations? 4 5 6 4 6 3 4 5 6 7 2 3 4 5 4 5 4 5 3 4 5 6 1 2 3 and β = 3 5 4 6 6 1 2 4 4 5 3 6 5 . Compute the following: (a) α−1 (b) αβ (c) βα 3. (a) What is the order of S6 (the symmetric group on 6 letters)? (b) How many elements of S6 are odd permutations? (c) Is the permutation (1356)(247) even or odd? 4. In S3 , find elements α and β such that |α| = 2, |β| = 2, and |αβ| = 3. 5. Find an isomorphism from the group of integers under addition, (Z, +), to the group of even integers under addition, (2Z, +). Be sure to show that it is an isomorphism (one-to-one, onto, φ(a + b) = φ(a) + φ(b). √ 6. Show that the mapping φ(x) = x from the group of positive real numbers with multiplication, (R+ , ∗), to itself is an isomorphism. (An isomorphism from a group to itself is called an automorphism.) 7. Let H = {0, ±3, ±6, ±9, . . . }. Find all cosets of H in Z. 8. Find the cosets of {1, 11} in U (30). 9. Suppose K is a proper subgroup of H and H is a proper subgroup of G. If |K| = 42 and |G| = 420, what are the possible orders of H? NOTE: Much of this is from the 6th edition of Gallian’s Contemporary Abstract Algebra. 1