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Math 366–001 HW 4, Spring 2015 This assignment is due Friday, March 13 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 subgroup 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. 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) √ 4. 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) 5. 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? 1 6. 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? 7. Let G and H be finite Abelian multiplicative groups. (a) What is the identity of G ⊕ H? (b) What is the inverse of element (g, h) ∈ G ⊕ H? (c) Show that G ⊕ H is also Abelian. 8. Prove that a quotient group of an Abelian group is Abelian. In particular, suppose G is a group with normal subgroup H. Prove that G/H is Abelian. (It might be useful to look up the group operation on G/H, since that’s what you want to show is commutative.) 9. What is the order of the additive group (Z ⊕ Z)/ h(2, 2)i? (h(2, 2)i is a normal subgroup of Z ⊕ Z since Z ⊕ Z is Abelian.) 10. If φ is a homomorphism from Z30 onto a group of order 5, determine the kernel of φ. (I want the actual kernel, not just its order!) 2