Math 455: Abstract Algebra I - Fall 2006

Math 455: Abstract Algebra I - Fall 2006



Important Notes


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Final Grades:

The total grades for the course were computed. The average for the total grades was 65, the median 64.14.

Here is how the grades were assigned and the number of people in each range:

Grade Average # of people
A 78-100 3
B+ 75-77 0
B 70-74 2
C+ 60-69 5
C 48-59 1
D 34-47 1
F < 34 0

You can round your average to the closest integer: 77.2 is a 77, 77.5 is a 78.

This is as nice as I can be! I've talked to a few professors, and this is the best I can do. As some of you might notice, I dropped the "you need at least a 20 in your In Class Final to get a C" requirement.

If you want to see your final or check your average, make an appointment.

I have already submitted the grades to the registrar office. I suppoese that you can already check it at Circle Park Online. I also posted them at Blackboard.


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Instructor Contact and General Information


Instructor: Luís Finotti

Office: Ayres Hall 212-D

Phone: 974-1321 (please do not ask me to call back -- leave your e-mail)


Office Hours: MWF 9:00am-10:00am or by appointment (subject to change!!)


Textbook: M. Artin. ``Algebra'', 1st Edition. Prentice Hall, 1991.

Prerequisite: Math 300 and Math 251 or 257.

Class: MWF 10:10-11:00 at Ayres Hall 214. (Section 1.)

Exams: Midterms: 09/29 (Fri) and 11/06 (Mon), in regular classroom and time; Final: 12/08 (Fri) from 10:15am to 12:15pm, also in our regular classroom.

Grade: 20% for HW + 20% for each Midterm + 40% for the Final. Note the weight of the HWs!


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Midterm I

Here is the Midterm I and the Midterm I with solutions.

The exams were graded and will be returned in class on Monday (10/02). If you are not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.) The average was 70.36 and the median was 74.5, which seems OK. The highest grade was 92.

Here is how I would curve the grades for this exam and the number of people with each grade:

Grade Range # of people
A 87-100 3
B+ 83-86 2
B 78-82 1
C+ 70-77 3
C 60-69 1
D 47-59 3
F < 47 1

Note: This curve is just to give you an idea of how I curve. In the end I will not even look at this curve, I will just curve all the final averages.


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Midterm II

You can check the Midterm II and the Midterm II with solutions.

The exams were graded and will be returned in the class Wednesday (11/08). The average and median were around 39, which is, of course, just horrible. The highest score was 77.

I will not give a curve to this exam here, since I can't make myself give a C to someone with 39. This means that we need to figure something out to settle this situation. We will discuss it in class on Wednesday.

Here is a little about the distribution of the grades.

Range # of people
75-100 1
60-74 1
45-59 2
35-44 4
25-34 3
20-24 1
< 20 1

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Makeup Midterm II

You can check the Makeup Midterm II and the Makeup Midterm II with solutions.

The exams were graded and were be returned in the class today, Wednesday 11/15. The average was 77.6 and the median 85. The highest score was 99. The results were good, but it was a simple exam, very similar to the Midterm II itself.

I will not give a curve to this exam, since I will curve only your ``official grade''. (See below.) But here is a little about the distribution of the grades.

Range # of people
90-100 3
85-89 3
80-84 1
75-79 2
70-74 1
60-69 0
< 60 2

Your ``Official Midterm II Grade'', which will count as 20% of your grade, replacing your actual ``Midterm II Grade'', was computed as

where G1 is your grade for Midterm II and G2 is your grade for the Makeup Midterm II. (Note: This gives the same result as the formula ``Max(G1,(1.5*Max(G1,G2)+Min(G1,G2))/2.5)'' which I had given before.)

The average of these grades was 59 and median was 64.4. The highest score is a 88.4.

Here is how I would curve these grades and the number of people with each grade:

Grade Range # of people
A 80-100 2
B+ 73-79 1
B 67-72 3
C+ 63-66 2
C 58-62 1
D 50-57 1
F < 50 2

Note: This curve is just to give you an idea of how I curve. In the end I will not even look at this curve, I will just curve all the final averages.


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Here are the finals: Take Home Final, Take Home Final Solution, In Class Final, In Class Final Solution.

The finals were graded. We had:

Exam Average Median Highest Score
Take Home 38.25 40.5 56
In Class 23.08 23.5 38
Final (Total) 64.03 65 87

The results of the Take Home part were reasonable, but the results of the In Class were pretty bad. Given the level of diffculty of the exam, and the amount of time given to you for the In Class, I find that pretty disturbing. We had 5 people below 20, and, as stated in class, those should not get a C in the course. (I will think about it and discuss it with others though.)

Here is the grade distribution for the Final (Total):

Range # of people
85-100 1
80-84 1
70-79 2
60-69 3
50-59 2
40-49 1
< 40 2

Note that the finals will not be returned, since they have to be kept on file. If you want to see your final or check your average, make an appointment.

There is no point in giving a curve for this grades, since I will curve your averages directly.


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Course Information:


In this course (and in its sequel, Math 456) you will be introduced to algebraic structures such as groups, rings, and fields. It is very important that you already have some practice with writing proofs (which was covered in Math 300) and familiarity with matrices and vector spaces (which was covered in Math 251 or 257). You will have to write many proofs in this course, and many of the examples we will look into will be related to matrices. Therefore, I will assume you have the proper background from both Math 300 and Math 251/257. If you had difficulty in those courses, you might need to review the material. (I would gladly try to help you with that if you come to my office hours.)


Homeworks will be assigned after every class and will be posted at this page. (See Homeworks below.) No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often!

The HWs will be collected on Mondays (except on Labor's Day week, when it will be due on Wednesday). (EDIT: HWs are now due on Wednesdays.) Each HW will have problems from the previous week (Monday, Wednesday and Friday lectures). The problems to be turned in, as well as due dates, will be clearly posted at the Homework section below. Note that not all of the problems turned in will be graded, but you won't know which until you get them back.

No late HWs will be accepted, except in extraordinary circumstances which are properly documented.

It is your responsibility to keep all your graded HWs and Midterms! It is very important to have them in case there is any problem with your grade.

I will do my best to post solutions. Please check the Important Notes section soon.

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is equal to the weight of a single midterm, and I will assume that you will work very hard on them.

Also, you should try to come to my office hours if you are having difficulties with the course. I will do my best to help you. Please try to come during my scheduled office hours, but feel free to make an appointment if that would be impossible.


You will have to check your e-mail at least once a week, preferably daily. I will use your e-mail (given to me by the registrar's office) to make announcements. If that is not your preferred address, write me an e-mail letting me know ASAP. I will assume that any message that I sent via e-mail will be read in a week or less, and it will be considered an official communication.

Due to privacy issues, I cannot send grades via e-mail, unless you sign a document saying that you are aware that e-mails are not secure and not necessarily private. If you want to sign such document, please let me know. (I will post a form at this page soon. Please check the Important Notes section soon.)


I have an Online Feedback Form where you can anonymously send me your comments and suggestions. I will consider your comments and try to do whatever I can to resolve possible problems before it is too late. So, please, feel free to use it whenever you have any constructive comment or suggestion. (In fact, I would greatly appreciate it.) If you don't want you comments to be anonymous, just send me an e-mail or come by my office and we can discuss the problem.

Course Structure -- UPDATED 12/04

We will likely cover the following portions from the text:

Chapters 2, (parts of) 5, and 6 give an introduction to basic concepts of ``Group Theory''. (Chapter 5 also deals with the particular case of ``Groups of Symmetries'' in details.) We will likely start Chapter 10, which deals with ``rings'', but we might have to finish it in Math 456.

Additional Bibliography

Here are some other books you might find helpful:

The first two books are considered ``easier'' books, and although they also have a somewhat different approach, they have most of the topics we will cover and may be of good help if you have difficulty reading Artin's book.

The last one is a ``standard'' text for a first course in abstract algebra, but might have a higher level of difficulty. (It's been used for the honors section of this course.) Nevertheless, it is a very good reference.  

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Legal Issues


All students should be familiar and maintain their ``Academic Integrity'': from
Hilltopics 2006/2007, pg.40:

Academic Integrity

The responsibility for learning is an individual matter. Study, preparation and presentation should involve at all times the student's own work, unless it has been clearly specified that work is to be a team effort. Academic honesty requires that all work presented be the student's own work, not only on tests, but in themes, papers, homework, and class presentation. There is a clear distinction between learning new ideas and presenting them as facts or as answers, and presenting them as one's own ideas. It is part of the learning process to incorporate the thoughts or ideas of others into one's own mind and presentations with the purpose of learning and enlarging on personal boundaries of knowledge.

You should also be familiar with the ``Classroom Behavior Expectations''.


Students with disabilities that need special accommodations should contact the ``Office of Disability Services'' and bring me the appropriate letter/forms.

Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the ``Office of Equity and Diversity''.  

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Important dates:


Wednesday, August 23 - Classes begin.

Friday, September 01 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, September 04 - Labor Day Holiday. (No class.)

Friday, September 29 - Midterm I.

Tuesday, October 03 - Last day to drop a full term course with a "W".

Thursday-Friday, October 12-13 - Fall Break. (No class.)

Monday, November 06 - Midterm II.

Tuesday, November 14 - Last day to drop with a WP/WF.

Thursday-Friday, November 23-24 - Thanksgiving. (No class.)

Tuesday, December 05 - Last Class Day.

Friday, December 08 - Final.


More dates.


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Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the ``answer''. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the answers. Be careful that just because our ``answers'' were the same, it doesn't mean that you solved the problem correctly (it might have been a ``fortunate'' coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions, though.

Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.

Solutions to Selected HW Problems (Click on ``Refresh'' or ``Reload'' if you don't see the changes!)


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HW1 - Due on Monday 08/28:

Review matrices and vector spaces (from Math 251/257).

Section 1.1 (pg. 31): 1, 2, 4, 11.

Section 1.2: 2, 12, 14.

Section 1.3: 1(b), (c), (d), 2 (Hint: row operations).


HW2 - Due on Wednesday 09/06:

Section 1.4: 1, 4.

Section 2.1 (pg. 69): 1(b), 3, 4, 7, 11. (Note: I did not do in class anything very similar to number 4, but just remember that every element of a group has an inverse. It is almost like you can ``divide'' both sides by something, but in fact you multiply both sides by an inverse to get rid of a term. But be careful: the group might not be Abelian!


HW3 - POSTPONED!!! (See Important Notes above.) Due on Wednesday 09/13:

Section 2.2: 2, 3, 4, 11, 13, 16(a), 17. Note: I excluded number 10 (from my previous list here)!


HW4 - POSTPONED!!! (See Important Notes above.) Due on Wednesday 09/20:

Section 2.3: 1, 4, 5, 9, 12.

Note: For 12 you need to know what an automorphism is: an automorphism is just an isomorphism from a group G onto itself.


HW5 - Due on Wednesday 09/27:

Section 2.4: 4, 8(b), 9, 13(a), 14, 17, 22.


HW6 - Due on Wednesday 10/04:

Section 2.5: 2, 5, 6. (Maybe you should also review a little about equivalence relations from Math 300.)


HW7 - Due on Wednesday 10/11:

Section 2.6: 1, 3, 5, 10(a). (Hint for 10(a): use Proposition 6.18)

Section 2.7: 1, 3.


HW8 - Due on Wednesday 10/18:

Section 2.8: 2, 3, 4(a), 4(b), 5, 6.


HW9 - Due on Wednesday 10/25:

Section 2.9: 1, 2, 5.

Section 2.10: 1(a), (b), (d), 3, 9, 10. (The last one was supposed to be 11.)


HW10 - Due on Wednesday 11/01:

Section 5.5 (pg. 192): 1(b), (c), 3, 8 (hint for 8(a)), 11(a).

Section 5.6: 1.

You can also try the Extra Credit Problem, but it is due in class Monday (10/30), NOT Wednesday!!! You are not allowed to discuss this problem with anyone!


HW11 - Due on Wednesday 11/08:

Section 5.6: 2, 6, 7.

Section 5.7: 6. (Hints: For (a) use problem 5: you can assume its statement is true without proving it. For (b), don't make it complicated. Look at some of our ``most used'' groups.)


HW12 - Due on Wednesday 11/15:

Section 6.1 (pg. 229): 1, 2, 4, 6, 10(a), (e), 12.

Section 6.3: 4, 5, 7, 13.


HW13 - Due on Wednesday 11/22:

Section 6.4: 1, 2, 3, 13, 15.


HW14 - Due on Wednesday 11/29:

Section 5.3 (pg. 189): 1, 2, 4. (These are about the dihedral group. Note that the text uses D_n for what we use D_2n.) Hint for 4(c): The answer is yes and to prove it you just need to show that the product group has two generators that satisfy the same properties as the two generators of the dihedral group.

Section 6.7 (pg. 233): 1.

Section 6.8: 1 (to show this, you basically need to show that you can get a and b from the two elements given in the problem), 6 (there is a typo in the statement: it should read ``N and G/N cyclic'' instead of ``G and G/N cyclic''), 16.

Extra Credit: Prove that if a group has order 24, then it is not simple. Alternatively, prove the same for order 36. (You can do either one.) These might be difficult... (Or, maybe I just did not see the easy way...) You are not allowed to discuss this problem with anyone! You should not look for the solved problem in other references either!


HW15 - not to turn in (just to study for the final):

Section 10.1 (pg. 379): 6 (you can replace the complex numbers with real numbers in this problem), 8(a), (c), 9(a), (c), 11(a), (b), 13.

Section 10.3: NOT IN THE EXAM! (Do it only if you want to have some fun...) 2, 4, 8(a), 9, 15, 17, 19, 20, 22, 29, 30.


PLEASE, HIT ``REFRESH'' (OR ``RELOAD'') IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.


If it is already Friday afternoon and there still is a ``More to come'' after the HW assignment due on the coming Monday, write me an e-mail at, and I'll update it and let you know.


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