Math 305: Introduction to Abstract Algebra and Number Theory

Assignments

FALL 2017

01:30 - 02:45 pm MW MB 139

Assignments must be handed in at the beginning of class on the due date.

This is an upper division class and you are expected to develop as an independent, intellectually honest scholar. Collected assignments is one of the instruments to assess your progress in this direction. Not every item covered in class is featured in the hand-in assignments, and not every item featured on the hand-in assignments is covered in class.

One of the learning outcomes of the course is to write clearly for specific purposes and audiences. This specific outcome will be fostered through the standards I will apply to collected work (homework assignments, examinations, projects). A problem solution is an expository work which explains in full detail, step-by-step, to anyone with access to the textbook for the course  how to obtain a solution to the problem. Note that if you were to find that a problem is not well-defined, or requires proving a statement that is false, you are expected to point this out in your work on the problem, providing the details that substantiate your findings.

You may discuss the homework problems with fellow enrollees in this course only, and you may consult any published literature on the subject. However, you must present your solutions in your own words, and you must properly attribute ideas and information gained from other sources. For the purposes of the homework assignments it is more important to produce a scholarly document that demonstrates your grasp of the material and your ability to properly and
thoroughly explain the solutions to these problems, than it is to be the inventor of these solutions. Answers without accompanying exposition that shows that you can coherently and logically explain the mathematical reasons justifying your answers will not receive any credit.

Content requirements for homework assignments:
a. The exposition of a solution must contain appropriate prose.
b. The exposition of a solution must, where appropriate, identify the mathematical steps taken, in logical order.
c. The exposition of a solution must properly reference the mathematical facts.

Format requirements for homework assignments:

• Your hand-in should be a typewritten document with at least 11 point font and margins of at least 1 inch on each side. Handwritten documents will not be accepted for grading, nor will partially handwritten work be graded. Prepare your final documents in time for the due time and date. Typesetting your documents in LaTeX is required.
• The upper left-hand corner of the first page of your document must contain in this order:
The homework set number
The due date of the homework
• A multiple page document must be stapled in the top left corner.
• The problems must be presented in the order they are assigned.
• Each problem solution must begin with the full statement of the original problem.
• There must be a visible separation between problems.

Format requirements are also applied at a number of other universities. See for example the requirements at Harvey Mudd College. You may want to use their LaTex homework template for yours.

Specific requirements:
The nature of some of the assignments may induce additional content and format requirements specific to that assignment. For these assignments the additional specific requirements will be stated with the assignment.

Policy:
1) No late hand ins will be accepted. Hand in on time, or take a zero for the assignment.
2) Hand ins not meeting ALL the format and specific requirements above will not be graded. Instead a zero will be recorded for the assignment.

1. Assignment 1 (Due September 13) Pinter: Page 29, Exercise A. [Possible: 16, Min: 6, Av: 13.45, High: 21]
2. Assignment 2 (Due September 25: Postponed from Sept. 20) Recall that $$F_n$$ denotes the $$n^{th}$$ Fibonacci number. These are defined by: $$F_1 = 1 = F_2$$ and for $$n>1$$, $$F_{n+1} = F_n + F_{n-1}$$. Prove that $$F_n$$ is even if, and only if, $$n$$ is divisible by 3.
[Possible: 13, Min: 7, Av: 11.5, High: 14]
3. Assignment 3 (Due October 02) (Based on U. Dudley's text) Define the prime counting function, $$\pi$$, so that for any positive integer $$x$$, $$\pi(x)$$ is the number of positive prime numbers no larger than $$x$$. Thus, $$\pi(1) = 0$$, $$\pi(2) = 1$$, $$\pi(3)=2$$, $$\pi(4)=2$$, and so on. In 1511 one Charles de Bovilles claimed that for all positive integers $$n$$, at least one of $$6n-1$$ and $$6n+1$$ is a prime number.
1. Determine the value of $$\pi(6n+1)$$ for all $$n\le 100$$.
2. Assuming De Bovilles is correct, what is the lower bound on $$\pi(6n+1)$$ for all $$n$$?
3. On the same axis system, sketch the graphs of $$\pi(6n+1)$$ and the lowerbound determined in the prior part. (Note that with value $$n$$ on the $$x$$-axis, we associate the value $$\pi(6n+1)$$, and the lower bound for this number based on de Bovilles' conjecture.)
4. Considering the graphs given in the previous part, is de Bovilles' Conjecture correct?
5. Based on actual data, is de Bovilles' Conjecture correct? Substantiate your answer with explicit evidence.
[Possible: 30, Min: 10, Av: 25.45, High: 34]
4. Assignment 4 (Due October 09) For a $$2\times 2$$ matrix $$\left(\begin{matrix} a & b \\ c& d\end{matrix}\right)$$ with entries in the set $$\mathbb{Z}_5 = \{0,\; 1,\; 2,\; 3,\; 4\}$$ define
$$det\left(\begin{matrix} a & b \\ c& d\end{matrix}\right) = ad-bc \mod 5$$
to be the $${determinant}$$ of the matrix. Let $$G$$ to be the set of such $$2\times 2$$ matrices with entries from $$\mathbb{Z}_5$$, and with determinant 1. Define the operation $$\odot$$ on $$G$$ to be matrix multiplication. Thus:
$$\left(\begin{matrix} a_1 & b_1 \\ c_1 & d_1 \end{matrix}\right) \odot \left(\begin{matrix} a_2 & b_2 \\ c_2 & d_2 \end{matrix}\right) = \left(\begin{matrix} a_1a_2 + b_1c_2 \mod 5 & a_1b_2 + b_1d_2 \mod 5 \\ c_1a_2 + d_1c_2 \mod 5 & c_1b_2 +d_1d_2 \mod 5\end{matrix}\right)$$
1. Determine if $$(G,\odot)$$ is a group.
2. Determine of the operation $$\odot$$ is commutative.
3. Determine the number of elements of the set $$G$$.
[Possible: 30, Min: 8, Av: 20.3, High: 38]
5. Assignment 5 (Due October 16) Consider a finite group $$(G,\odot)$$ for which $$\vert G\vert = pq$$ where $$p < q$$ are odd prime numbers. Assume that the group is Abelian. Either prove that the group is cyclic (i.e. has a generator), or demonstrate that the group need not be cyclic by providing a counter-example.
Solution by a class-mate.
[Possible: 15, Min: 6, Av: 12.2, High: 16]
6. Assignment 6 (Due November 08) For this assignment you may team up with up to two other people enrolled in the class to collaborate on the assignment and hand in a joint paper. It is not required to team up with anybody. You may submit an individual assignment if you prefer to do so.
Consider the symmetric group $$(\textsf{S}_{9},\circ)$$.
1. Determine the number of factors of the order of this group.
2. Determine if $$(\textsf{S}_9,\circ)$$ has a subgroup of order 35.
3. Determine if $$(\textsf{S}_9,\circ)$$ has a subgroup of order 21.
7. Assignment 7 (Due November 29) For this assignment you may team up with up to two other people enrolled in the class to collaborate on the assignment and hand in a joint paper. It is not required to team up with anybody. You may submit an individual assignment if you prefer to do so.
Consider groups of the form $$(\textsf{U}_{n}, * \mod n)$$.
1. For $$n=303$$:
• Determine the number of factors of the order of this group.
• For each factor of the order of this group, determine the number of subgroups of that order.
• Determine the largest order of a cyclic subgroup of this group.
2. For $$n=3*p$$ where $$p$$ is a prime number:
• Determine if there is a maximum number of factors of the order of groups of this form.
• Determine if there is a maximum number of cyclic subgroups of a given order dividing the order of a group of this form.