Abstract Algebra I

Fall  04


 First Day Handout

Homework Exercises

Gallian Website

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Abstract Alg. Applications

Class Notes



12/13/04 New Final Exam date and location:  1:00 to 3:30, Tuesday, December 14, in JEM 376 (a comfortable seminar room).
11-10-04 The final exam will be from 9:00 to 11:30 on Monday, December 13.  Class will meet at Libby's at 8:45 on Friday, December 10, to discuss the three problems.
11-8-04 Midterm revisions:  On Friday, Nov 12, please bring to class your individual problems to turn into me, and a photocopy of them to keep for yourself.  We will put these problems up on the board.  Over the weekend, please collaborate with one another to write up ALL of the problems beautifully.  On Monday, Nov 15, please turn in these beautiful write ups, together with the problems from the original midterm.
11-1-04 RSA Code information and due dates here: RSA CODE.  Here are directions for connecting to Maple from your room:  http://academics.smcvt.edu/jellis-monaghan/calc1/maple%20labs/Maple_from_room.htm


10-27-04 PowerPoints from the required talk is at: Mary Cox Talk

Get the video from me.

10-19-04 Required Talk--

Date:  Tuesday, October 26, 2004

Time:  3:30 – 4:30 pm

Location:  Jeanmarie 391, SMC Campus.


Title:  The Continued Fraction in Combinatorial Game Theory


Speaker:  Mary Cox, UVM.





The origin of continued fractions is traditionally placed at the time of the creation of Euclid's Algorithm; in fact, continued fractions are intimately related to Euclid's Algorithm.  In the 19th Century, there was an explosion of growth within this subject, with contributions by Jacobi, Perron, Hermite, Gauss, Cauchy, and Stieljes.  Since the beginning of the 20th Century continued fractions have made their appearance in other fields, including chaos theory, within computer algorithms for computing rational approximations to real numbers, solving indeterminate equations, and knot theory.  Very recently, continued fractions have also been used in the relatively young field of combinatorial game theory.


In this talk, we define the general and simple continued fraction, and give a short demonstration of a traditional number theoretic use -- solving Pell's equation.  We then outline the basics of combinatorial game theory, and then demonstrate the use of continued fractions in this area of mathematics by demonstrating the evaluation of a game of Contorted Fractions.


This talk is accessible to a general audience.




As part of the President's Distinguished Lecture Series the Departmentof Mathematics and Statistics is pleased to host Ronald Graham.

He will be speaking at

4:00pm Monday October 18th in Angell B106 on the UVM campus.

A reception will follow in Billing's Apse.

Dr. Graham is the current president of the Mathematical Association of America,past president of the American Mathematical Society, and of the International Jugglers Association. He is in the Guinness Book of World Records for the largest number ever used in a mathematical proof,and in Ripley's Believe it or Not for his skills as a trampolinist and juggler.

For more details on Dr. Graham see: http://www.math.ucsd.edu/~fan/ron/

Title: Searching for the shortest network

Abstract: There are many situations in which one would like to connect a collection of points together by a network having the minimum possible total length. Such problems occur in the design of telephone networks, railroad lines, oil and gas pipeline networks, heating and air-conditioning duct systems, the layout of VLSI chips, and the inference of evolutionary pathways for living organisms, for example. In this talk we give a summary of what is known (and unknown) about this classical problem, and how current developments in computer science have impacted it.

If you only go to one math talk this year, this should be it!

There is also a second, earlier, talk:

Archimedes, Combinatorics and the Stomachion

Professor Ron Graham

University of California at San Diego

Monday, October 18, 2004

2:30-3:20 PM

Room L200 Lafayette
Reception to follow in Billings Apse at 5:00pm


Greek mathematics has long been known for its fundamental contributions to geometry and diophantine equations, for example  However, there is now evidence that ancient Greek mathematicians were also interested in various combinatorial problems. In this talk, I will describe some recent work exploring this theme which arose from the analysis of a puzzle called the Stomachion, which was recently discovered in a long lost manuscript of Archimedes.


For extra credit on the next test, attend one or both of the talks and write about 2 or 3 paragraphs describing the talk.

Note--if you absolutely cannot make either of these talks, but would like to take advantage of the extra credit, we have a videotape of one of Ron Graham's lectures at the library, and you can base your write up on that.  Call Number: VTC 2421 to get the video from the library.


MIDTERM:  Take home part. 

1.  Redo the Three Examples, using the theory you have learned as well as your own intelligence.   You may discuss these with one another as much as you like (in fact I strongly recommend this), however, each person must do their own write-up!!


2. Prepare Study Sheets for chapters 0 through 6. You will be able to use these on the Final Exam (although not on the Midterm), so the more thorough, the better.  They should include all theorems and definitions, with small examples attached to help you understand what is being said.  There is no page limit here, but if it is longer than 3 or 4 pages, you might want to come up with a system to let yourself find things easily during a test.


3.  Write up the following exercises beautifully--presentation and clarity count!  pg 89-92, numbers 18, 24, 26, 32. pg 169-171 numbers 2, 6, 48, 52 (need that n>2).  You must work entirely on your own--no collaborating or discussing these problems in any way with anyone other than myself.

Take home part is due October 20 at the beginning of class.

Inclass part:  This will consist of true/false from the Gallian website, or very similar,  (to check your understanding of the theory), as well as basic computation examples.  Eg you should be able to find (quickly) gdcs, lcms, s and t for linear combinations, orders of various elements in various groups, subgroup lattices, permutations (multiply, invert, write as transpositions, find orders, know Sn and An), group tables.  This portion is really to see that you are adept at the basic computation necessary to work small examples.  The inclass test will be on Wednesday October 20.


?? We agreed to do the exercise at the bottom of chapters 3 and 4 at: http://www.mapleapps.com/powertools/abstractalgebra/abstractalgebra.shtml     (due wednesday Oct 6)  Also here are some examples of using Maple to multiply permuations: permutation group.mws
9/24/04 We agreed to do the exercise at the bottom of chapter 2 at: http://www.mapleapps.com/powertools/abstractalgebra/abstractalgebra.shtml     (due wednesday)
9/22/04 Putnam Competition will be held this year (as every year) on the first Saturday in December, and we are looking for students interested in possibly being a part of the competition.  See Lloyd Simons if you would like to participate.
9/22/04 From George Ashline

I wish to bring to your attention that once again this year there will be a conference for undergraduate students interested in mathematics during the Spring semester. The twelfth annual Hudson River Undergraduate Mathematics Conference (HRUMC) will take place on Saturday, April 30, 2005 at Williams College in Williamstown, Massachusetts. The keynote speaker will be Ken Ribet of UC Berkeley. In the past several years, a contingent of students and faculty from St. Michael's has gone to the HRUMC, and several students and faculty members have given presentations at previous conferences. For more information about this meeting, you can check out the conference homepage at http://www.skidmore.edu/academics/mcs/hrumc.htm. At this site, you can find details about last year's conference which was held at Mount Holyoke College.

At the HRUMC, some talks are given on original research in mathematics or a related field (done by students, faculty, or jointly), while others are expositions of interesting work that has been accomplished and discoveries that have been made in mathematics (typically not discussed in regular course work). If you think that you might be interested in giving a talk or attending the conference to see what it's like, please let me or anyone else in the mathematics department know. If you'd like to get more information about this opportunity, just e-mail me back or stop by my office and I'd be happy to discuss it more with you. I have copies of recent conference programs if you'd like to get a feel for what the conference involves and the type of talks given. I look forward to hearing back from some of you about this opportunity.


Dear Math Students,

On Saturday, September 25 the Math Department will hold a very special event – our own celebration of the Centennial.

There will be a talk, an honor society induction and a Career Panel – as well as birthday cake to celebrate 100 years of Math at SMC.

A number of Math alums are coming from far and wide to connect with you, our current students – who will follow in their footsteps.

This is a great opportunity to network with alums working in a variety of areas (see below!) – and to learn about applications of math.

This is really an exciting program that should be of special interest to you --let’s have a great turnout on September 25th!

And please invite your friends -- and family if they are visiting.

Zsu Kadas and the Math Faculty


Academic Internships!

Academic interships are a great way for students to strengthen their résumés, increase their marketability and test the career waters.

The academic intership office holds informational meetings in September for Spring internships and in January for Fall internships. This semester’s meetings will be held on September 20, 21, 22, and 23 from 11:45 – 12:15 in Jeanmarie 376. Joanne LaBrake is available in Klein 114, extension 2314 if you have any questions.

9/3/04 http://www.cvfair.com/   A link to the fair!
8/27/04 You may order a personal copy of Maple 9.5 at a discounted price of $75 if you wish:  Secure ordering address: http://webstore.maplesoft.com Promotion Code: AD1118


Some career resources:

Lots of people with exciting real jobs using Math, More people using Math,  Job Listing Services, Comprehensive resources from SIAM AWM Career resource site.

The department has (in George's office) a CD-ROM and videotape entitled Careers in Mathematics created by three major professional societies of mathematicians (AMS, MAA, and SIAM).  He also has video tapes of a panel and presentation by recent SMC math alums.

There is also this note from George:

"This summer I have been in contact with a number of St. Michael's mathematics alumni in a wide variety of careers, in such areas as statistical data analysis, actuarial consulting, software engineering and other computer-related fields, education (at the secondary and university levels), and even flying (specifically as a USAF pilot). These alumni are willing to be resource/contact people representing a variety of careers. If you wish to be in contact with an SMC alumnus from a particular mathematics-related field, then let me know and I will try to connect you with an appropriate person. I recently have given a few of my advisees e-mail addresses of mathematics alumni who are pursuing careers in an area of interest to them, and could do a similar service for any of you that are interested."



A note from math dept chair Jim H:

Welcome back and hello,

For many years we've run math help sessions at night. These provide a chance for students in calculus classes, or in the 101 and 102 classes, to get somy help with their material on a drop-in basis. These sessions are staffed by students, so this is where you come in!

Are you interested in working one or two? We always pair people up, so you won't be lonely, and we get many people who have done it before that come back again, so it must be fun. If you have had through Calc III, and would be interested in making a little extra money (I am told, about $7-$8 per hour), working one or two nights in a one-on-one or small group setting, then I'd like to hear from you. Tell me what classes you've had, please, and what nights you could work.

Also, we are looking for folks who might be interested in grading homeworks or quizzes. You work with a faculty member, again in conjunction with a calculus, or 101, or 102 class. You will be grading quizzes or homework papers, following the instructions of the course instructor. If you are interested in this, please also write and tell me which classes you've had.

Thanks, Jim Hefferonjim@joshua.smcvt.edu 


Be sure to click 'refresh' when you visit this site to be sure you get the most recent changes.


The Student Resource Center runs excellent workshops every semester on study skills, time management, test-taking techniques, etc.  For details: SRC study skills  


The postings on this site have all been tested and all work.  If you have any difficulty getting material off this site, you can try any of the following:

1.  See if someone in your study group was able to print the file you want and make a photocopy of it.

2.  Try using one of the lab computers in case the problem is with how your own computer is configured.

3.  Using Explorer, right click on the link, select "save target as", then save the file to your home computer.  You should be able to open and print it from there.

4.  Contact Cynthia Kelley at 654-2756 or ckelley2@smcvt.edu.  She is the person for website trouble shooting.

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