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Name: Problem Set 7 Math 415 Honors, Fall 2014 Due: Tuesday, November 11. Review Sections 16 in your textbook. Complete the following items, staple this page to the front of your work, and turn your assignment in at the beginning of class on Tuesday, November 11. Remember to fully justify all your answers, and provide complete details. Neatness is greatly appreciated. 1. Let X be a set, and G a group acting on X. Let Y ⊆ X. Let GY = {g ∈ G | g∗y = y for all y ∈ Y}. Show that GY is a subgroup of G. 2. Let G be the additive group of real numbers. We define an action of G on the set R2 by setting θ ∗ P to be the result of rotating P ∈ R2 counterclockwise about the origin through θ radians. a. Show that X = R2 is a G-set, that is, show that what I claimed to be an action is indeed an action. b. Let P ∈ X a point which is not the origin. Describe geometrically the orbit of P under the action of G. c. Let P ∈ X a point which is not the origin. Find the group G P . 3. Let G be a group and let X = {subgroups of G}. Show that G acts on X by conjugation. That is, show that g ∗ H := gHg−1 , for g ∈ G and H ∈ X, defines an action of G on X. 4. Let G be a group and let H be a subgroup of G. Show directly from the definition that {g ∈ G | gHg−1 = H} is a subgroup of G. For this problem, do not use actions. 5. Show that the set {identity, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)} ⊆ A4 is a normal subgroup of A4 , that is, you have to prove this set is a subgroup first, then prove it is normal. Conclusion: A4 is not a simple group. 6. Let a, b, c be distinct elements of {1, 2, 3, 4, 5}. Show that there exists an element σ ∈ A5 such that σ(1, 2, 3)σ−1 = (a, b, c). (This will give us a conjugacy class of cardinality 20 in A5 .) Hint: You have a choice of where to send 4 and 5. 7. Let a, b, c, d be distinct elements of {1, 2, 3, 4, 5}. Show that there exists an element σ ∈ A5 such that σ(1, 2)(3, 4)σ−1 = (a, b)(c, d). (This will give us a conjugacy class of cardinality 15 in A5 .) Hint: (a, b) = (b, a). 8. Consider the cycle (1, 2, 3, 4, 5) ∈ A5 . Find 11 other cycles of the same length that are conjugate to this one in A5 . Page 1 9. Consider the cycle (1, 3, 5, 2, 4) ∈ A5 . Find 11 other cycles of the same length that are conjugate to this one in A5 . 10. Show that (1, 2, 3, 4, 5) and (1, 3, 5, 2, 4) are not in the same conjugacy class in A5 . Together with the previous exercises, this shows that there are two conjugacy classes of cardinality 12 in A5 , corresponding to cycles of length 5. 11. Read Theorem 16.16 and its proof in the book. Write the proof in your own words. 1/10 2/10 3/10 4/10 5/10 6/10 7/10 8/10 9/10 10/10 11/10 Through the course of this assignment, I have followed the Aggie Code of Honor. An Aggie does not lie, cheat or steal or tolerate those who do. Signed: Page 2 Total