数学finalExam代考 Math 417: Exam 3 Instructions

Math 417: Exam 3

数学finalExam代考 For each point p0 = (x0, y0), the orbit R·p0 is a subset of R2 that turns out to have a simple description in terms of the

Due Friday, April 23, 2021

Instructions: 数学finalExam代考

• This is a take-home exam. Please solve the problems and upload your solutions to Moodle.


• The deadline to submit the exam is 11:59pm on Friday, April 23, 2021.


• Please submit your exam in PDF format if at all possible.


• There are 100 points possible on this exam.


• Unlike the homework, collaboration is not permitted on the exam. You are allowed to refer to the textbook, lecture notes, lecture videos, homework solutions on Moodle, your own home work, and any notes that you have made. You may not consult other resources when taking the exam.

• Any questions about the exam should be directed to your instructor.

Problems:  数学finalExam代考

1. (20 points) Construct a nonabelian group of order 55, and show that the group you construct is nonabelian by fifinding two elements that do not commute. Hint: Use a semidirect product of two cyclic groups.

2. Consider the group G = R (with addition), and the set X = R2 (the plane). There is an action of R on R2 given by r ·(x, y) = (x +r y, y)

(a) (5 points) Prove that this formula does indeed defifine a group action.

(b) (8 points) For each point p0 = (x0, y0), the orbit R·p0 is a subset of R2 that turns out to have a simple description in terms of the geometry of the plane. Find this geometric description of R·p0, and justify your answer. (The description depends which point p0 you start with.)

(c) (7 points) For each p0 = (x0, y0), fifind the stabilizer of p0, and justify your answer. 数学finalExam代考

3. (20 points) Find the number of necklaces that can be made with 6 beads of k colors, write your answer as a polynomial function of k, and justify your answer.

4. (15 points) Let G be a group such that |G| = 16 and |Z(G)| = 2. Prove that G must have a conju gacy class whose size is exactly 2.

5. Suppose that G is a group of order 30659 = 23 · 31 · 43.

(a) (10 points) Prove that, for each prime p dividing |G|, any p-Sylow subgroup is normal. (For this problem, you may use a calculator to check divisibility relations, but you should explain what calculations you performed.) 数学finalExam代考

(b) (10 points) Prove that G is isomorphic to a direct product of Sylow subgroups of itself.

(c) (5 points) Prove that G is cyclic.

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