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Math 366–002 HW 4, Spring 2014 This assignment is due Friday, March 7 in class. Feel free to work together, but be sure to write up your own solutions. As for writing it up, please write legibly on your own paper, including as much justification as seems necessary to get the point across. 1. Suppose G is a finite cyclic group of order 20, generated by some element a. Make a list of all possible orders of each subgroups of G, along with a generator for each, i.e., order 20 generator a .. . 2. Suppose G is a finite cyclic group that has exactly three subgroups: G, the trivial subgroup, and a subgroup of order p (where p is some prime number). What is |G|? (It might be most useful to think about specific cases first, like p = 5, but I want a general answer for any choice of p.) 3. Find a collection of distinct subgroups ha1 i , ha2 i , . . . , han i of Z240 with the property that ha1 i ⊂ ha2 i ⊂ . . . ⊂ han i with n as large as possible. 4. What is the order of each of the following permutations? 1 2 3 4 5 6 7 8 (a) 2 4 3 1 6 7 8 5 (b) (1 4 25 3)(32 11)(13 21 24 27 176) (c) (1 2)(1 3)(2 3) 5. Find an isomorphism φ from (Z, +) to (2Z, +) (the additive group of even integers). (a) Show that φ is onto by showing that any element 2k ∈ 2Z is the image of some element of Z. (b) Show that φ is one-to-one by showing φ(a) = φ(b) implies that a = b. (c) Show that φ preserves the group operation, i.e., φ(a + b) = φ(a) + φ(b). p 6. Consider the map ψ : R+ → R+ given by ψ(x) = (x). Show that ψ is an automorphism of the multiplicative group of positive real numbers, R+ . In particular: (a) Show that ψ is onto. (b) Show that ψ is one-to-one. (c) Show that ψ preserves the group operation, i.e., ψ(ab) = ψ(a)ψ(b) 1 7. Let G be an Abelian multiplicative group, and let α : G → G be given by α(g) = g −1 . Show that α is an automorphism of G (plus one extra thing – part (d) below!): (a) Show that α is onto. (b) Show that α is one-to-one. (c) Show that α preserves the group operation. (d) Why does α fail to be an automorphism if G is not Abelian? 8. Find all cosets of H = {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? 10. Let G be a group with |G| = pq (p and q prime). Prove that every proper subgroup of G is cyclic. 2