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Name: Problem Set 5 Math 416, Section 200, Spring 2014 Due: Tuesday, March 3rd. Review Sections 16 and 36 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, March 3rd. Remember to fully justify all your answers, and provide complete details. 1. Let X be a G set, and 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 make the set R2 into a G-set by defining 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 finite group. Show that G is a p-group if and only if |G| is a power of p. 4. 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. 5. 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. 6. Find all Sylow 3-subgroups of S 4 and verify that they are conjugate. 7. Read Example 36.13. Show that every group of order (35)3 has a normal subgroup of order 125. 8. Extra Credit. Show that there are no simple groups of order pr m, where p is prime, r is a positive integer, and m < p. Page 1 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