Abstract Algebra

Fall 2023

**8 December:**The assignment average and term score for each student in the class have been computed and posted on the Avenue To Learn site for this course. The assignment average is a grade out of 10 and was computed by weighing each of the 6 assignments equally, dropping the lowest weighted assignment score, and averaging the rest. The term score was obtained by adding the assignment average, the first midterm test score (scaled out of 25 points, by multiplying the raw test score by (25/23)), and the second midterm test score. The term score is out of 60. If you have any questions about either scores, please contact Dr. Valeriote as soon as possible. The average of the term scores for the class is 75%.**8 December:**Solutions to Assignment #6 have been posted here and the assignment has been graded. Feedback on your submission can be found on the Avenue To Learn site for this course. If you have any questions about the grading, please first carefully review the posted solutions and then contact the course TA Mike, or Dr. Valeriote, should you still have questions.**6 December:**Dr. Valeriote's office hour on Friday, December 8 will be held from 9:00am to 10:00am instead of at the previously announced time. The course TA, Mike, will hold an office hour on Tuesday, December 12 from 12:30 to 1:30pm in Hamilton Hall Room 312. Details on the final exam can be found here.**28 November:**The MATH 3GR3 lecture on Friday, December 1 will be given by the course TA Mike. He will lecture on material from Section 17.1 of the textbook. Office hours on Friday, December 1 for both Dr. Valeriote and Mike have been cancelled. Mike will hold an office hour on Tuesday, December 5 from 12:30 to 1:20pm in Hamilton Hall Room 312. Dr. Valeriote will hold his normally scheduled office hour on Tuesday December 5.**26 November:**Solutions to Assignment #5 have been posted here (LaTeX source) and the assignment has been graded. Feedback on your submission can be found on the Avenue To Learn site for this course. If you have any questions about the grading, please first carefully review the posted solutions and then contact the course TA Mike, or Dr. Valeriote, should you still have questions.**25 November:**The final exam for this course will be held on Tuesday, December 12 from 7:30pm to 10:00pm (2.5 hours in length). Details on the final exam can be found here. Check on Mosaic to find the location of the exam.**25 November:**Assignment #6 has been posted here. It is due on Wednesday, December 6 by 11:59pm. The LaTeX source for this assignment can be found here.**16 November:**The second midterm test has been graded and grades have been posted on the Avenue To Learn site for this course. You can find a copy of the solutions here. Tests will be returned during the tutorial on Tuesday, 21 November. After that date, they can be picked up at Dr. Valeriote's office, during office hours. Once you have obtained your test copy, please carefully read it over and check to see that your grade was correctly entered into Avenue To Learn. If you have any questions about how your test was graded, please first look over the posted solutions and then contact Dr. Valeriote. The test average and median are 75% and 78% respectively.**13 November:**Solutions to Assignment #4 have been posted here (LaTeX source) and the assignment has been graded. Feedback on your submission can be found on the Avenue To Learn site for this course. If you have any questions about the grading, please first carefully review the posted solutions and then contact the course TA Mike, or Dr. Valeriote, should you still have questions.**12 November:**Important Reminder: The second MATH 3GR3 midterm test will be written on Tuesday, 14 November from 11:30am to 12:20pm in the lecture hall where the MATH 3GR3 tutorials are held. Details on the test can be found here.**12 November:**Assignment #5 has been posted here. It is due on Friday, November 24 by 11:59pm. The LaTeX source for this assignment can be found here.**9 November:**Students who are unable to easily attend the MATH 3GR3 lectures due to the HSR strike, may view copies of the lectures (photographs of the blackboard in the lecture hall) online. Download links can be found in the table of lectures below. The first set of photographs are for today's lecture, lecture #26. They were provided by a generous student from the class.**2 November:**Details of the second midterm test can be found here. It will be written from 11:30am to 12:20pm on Tuesday, November 14 (the scheduled tutorial time for this course). It will be written in the scheduled tutorial room for this course. Please ensure that you know where this room is, well in advance of the test date. If you have any questions about the test, please contact Dr. Valeriote.**1 November:**Assignment #3 has been graded. Feedback on your submission can be found on the Avenue To Learn site for this course. If you have any questions about the grading, please first carefully review the posted solutions and then contact the course TA Mike, or Dr. Valeriote, should you still have questions. Solutions to assignment #3 have been posted here (LaTeX source here). When uploading solutions to homework assignments, please try to limit the file size of your document to less than 10MBytes. Larger files are difficult to process on Avenue To Learn.**27 October:**Assignment #4 has been posted here. It is due on Friday, November 10 by 11:59pm. The LaTeX source for this assignment can be found here.**20 October:**The first midterm test has been graded and grades have been posted on the Avenue To Learn site for this course. You can find a copy of the solutions here. Tests will be returned during the tutorial on Tuesday, 24 October. After that date, they can be picked up at Dr. Valeriote's office, during office hours. Once you have obtained your test copy, please carefully read it over and check to see that your grade was correctly entered into Avenue To Learn. If you have any questions about how your test was graded, please first look over the posted solutions and then contact Dr. Valeriote. The test was a little long and so Question 5 b) is now considered a bonus question, worth 2 marks. So, the test is out of 23, not 25. With this change, the test average and median are 75.7% and 78.2% respectively.**18 October:**An information session on graduate school will be held on Wednesday, October 25 at 5:00pm. Details on the session can be found here.**11 October:**Assignment #2 has been graded. Feedback on your submission can be found on the Avenue To Learn site for this course. If you have any questions about the grading, please first carefully review the posted solutions, (LaTeX source here) and then contact the course TA Mike, or Dr. Valeriote, should you still have questions.**6 October:**Assignment #3 has been posted here. It is due on Friday, October 27 by 11:59pm. The LaTeX source for this assignment can be found here.**2 October:**Details of the first midterm test can be found here. It will be written from 11:30am to 12:20pm on Tuesday, October 17 (the scheduled tutorial time for this course). It will be written in the scheduled tutorial room for this course. Please ensure that you know where this room is, well in advance of the test date. If you have any questions about the test, please contact Dr. Valeriote.**28 September:**A note on some connections between the order of a group and the order of an element in the group can be found here. It provides a more organized presentation of some facts that were discussed during the lecture on Tuesday, 26 September.**27 September:**Assignment #1 has been graded. Feedback on your submission can be found on the Avenue To Learn site for this course. If you have any questions about the grading, please first carefully review the posted solutions, and then contact the course TA Mike, or Dr. Valeriote, should you still have questions.**26 September:**Solutions to Assignment #1 can be found here (LaTeX source here).**25 September:**Assignment #2 has been posted here. It is due on Friday, October 6 by 11:59pm. The LaTeX source for this assignment can be found here.**12 September:**The course TA, Mike, will hold office hours on Thursdays from 12:30 to 1:20pm and Fridays from 1:30 to 2:20pm. Once rooms have been booked for these office hours, their locations will be posted on the Avenue To Learn site for this course.**11 September:**The first MATH 3GR3 tutorial will be held on Tuesday, September 12 at 11:30am. Check the Avenue To Learn site to find out its location, or send an email to Dr. Valeriote.**5 September:**A MATH 3GR3 Discussion Forum has been set up on the course Avenue to Learn site. Feel free to use it to pose questions related to the course, including questions about the homework. Dr. Valeriote or the course TA Mike, will try to respond to any questions in a timely manner. Please do not provide answers to any of the homework problems. The Discussion Forum can be found by clicking on the "Communication" pull down menu that is located near the top of the site.**5 September:**The TA for this course is Mike Cummings. He can be reached, via email, at cummim5@mcmaster.ca. The first tutorial of the semester will be held on Tuesday, September 12 from 11:30am to 12:20pm. Details on its location can be found on the Avenue To Learn course site or on Mosaic.

**5 September:**A record of the topics covered in each lecture, along with a tentative schedule for upcoming lectures, can be found near the bottom of this webpage (here). It will be updated on a regular basis.**4 September:**Read over the introductory material found in Chapters 1 and 2 of the course textbook this week.**23 August:**Assignment #1 has been posted here. It is due on Friday, September 22 by 11:59pm. The LaTeX source for this assignment can be found here. Please read over the instructions at the top of the assignment. In particular note that you will need to upload to the course Avenue To Learn site, a single pdf file that contains all of your solutions to this assignment. No other file formats will be accepted. If you need assistance with this, please contact the course TA or Dr. Valeriote prior to the assignment deadline.- There will be two midterm tests during the semester, with each one worth 25% of the final grade. The tests will be held, in person, during the scheduled tutorial time during the weeks of October 17 and November 14 (tentative). So, it is important that registered students be able to attend in person the tutorials, especially on the designated test dates. The test dates and other test details will be confirmed at the start of the semester.
- We will make use of the very powerful and useful computational package called SageMath. More details on this package can be found at the bottom of this webpage and also in the online version of the course textbook. It will also be used and demonstrated during the lectures. Here is a link to a SageMath tutorial: https://doc.sagemath.org/html/en/tutorial/
- The text for the course can be downloaded here, at no cost. We will use the 2022 annual edition of the book in this course, but any recent edition will work well. Since there will be a Sage component to this course, then consider downloading the 2022 edition of the text that includes Sage. It can be downloaded here.
- There will be a tutorial for this course. It will be held on Tuesdays from 11:30am to 12:20pm.
- Dr. Valeriote's office hours will be held from 9:45am to 11:15am on Tuesdays and Fridays. He is also available by appointment.
- The course announcement can be found here.
- The first class will be held on Tuesday, September 5 at
8:30am.

Assignments, Tests, and Solutions:

- Assignment #1 (LaTeX source) Solutions (LaTeX source)
- Assignment #2 (LaTeX source) Solutions (LaTeX source)
- Assignment #3 (LaTeX source) Solutions (LaTeX source)
- Assignment #4 (LaTeX source) Solutions (LaTeX source)
- Assignment #5 (LaTeX source) Solutions (LaTeX source)
- Assignment #6 (LaTeX source) Solutions
- Midterm Test #1 (Solutions)
- Midterm Test #2 (Solutions)

Lecture Schedule (tentative):

Lecture # | Date | topics covered | reading/resources/comments |

1 | 05/09/23 | Course introduction, equivalence relations | section 1.2 from textbook |

2 | 07/09/23 | equivalence relations, partitions | section 1.2 |

3 | 8/09/23 | partitions, division algorithm | sections 1.2, 2.2 |

4 | 12/09/23 | gcd(a, b), Euclidean algorithm, symmetries of the rectangle | sections 1.6, 3.1 |

5 | 14/09/23 | Cayley table of symmetry groups of rectangle and triangle, definition of a group |
sections 3.1, 3.2 |

6 | 15/09/23 | definition of a group, examples, basic properties: the
cancellation law for groups, properties of the Cayley table, exponentiation laws |
section 3.2 |

7 | 19/09/23 | group of integers modulo n, group of units modulo n | section 3.2 |

8 | 21/09/23 | subgroups, examples, Proposition 3.30 | sections 3.2, 3.3 |

9 | 22/09/23 | examples of subgroups, cyclic subgroups, and cyclic groups. Assignment #1 due. | sections 3.3, 4.1 |

10 | 26/09/23 | cyclic groups, order of groups, elements. Cyclic groups are abelian, every subgroup of a cyclic group is cyclic | sections 4.1 Some notes |

11 | 28/09/23 | cyclic groups, orders of elements in a cyclic group, permutation groups | sections 4.1, 5.1 |

12 | 29/09/23 | permutation groups, examples, cycles | section 5.1 |

13 | 03/10/23 | cycle decomposition of permutations, transpositions, examples | section 5.1 |

14 | 05/10/23 | even and odd permutations, the alternating groups | section 5.1 |

15 | 06/10/23 | Lemma 5.14, dihedral groups. Assignment #2 due. | section 5.2 |

16 | 17/10/23 | Midterm test review. Midterm
test #1 (during tutorial). |
sections 5.2, 6.1 |

17 | 19/10/23 | left and right cosets of a subgroup | section 6.1 |

18 | 20/10/23 | cosets, Lagrange's Theorem | sections 6.1, 6.2 |

19 | 24/10/23 | Lagrange's Theorem, Fermat's Little Theorem, Euler's Theorem | sections 6.2, 6.3 |

20 | 26/10/23 | isomorphisms | section 9.1 |

21 | 27/10/23 | isomorphisms, cyclic groups, Cayley's Theorem. Assignment #3 due. | section 9.1 |

22 | 31/10/23 | Cayley's Theorem, Cartesian Products | sections 9.1, 9.2 |

23 | 02/11/23 | Cartesian product, product of cyclic groups, internal direct product. | section 9.2 |

24 | 03/11/23 | internal direct product, normal subgroups | sections 9.2, 10.1 |

25 | 07/11/23 | factor groups, examples. | sections 10.1, 10.2 |

26 | 09/11/23 | simple groups, homomorphisms, examples, properties | section 11.1 Lecture Notes |

27 | 10/11/23 | the kernel, properties of the kernel, canonical homomorphism, First Isomorphism Theorem. Assignment #4 due. | sections 11.1, 11.2 Lecture Notes |

28 | 14/11/23 | midterm test review. Midterm test #2 (during tutorial). | |

29 | 16/11/23 | the first isomorphism theorem and applications. | section 11.2 Lecture Notes |

30 | 27/11/23 | Rings, ring examples, basic properties | section 16.1 |

31 | 21/11/23 | integral domains, ring homomorphisms | sections 16.2, 16.3 |

32 | 23/11/23 | homomorphisms, kernels, ideals. | section 16.3 |

33 | 24/11/23 | ideals, continued. Assignment #5 due. | section 16.3 |

34 | 28/11/23 | factor rings, canonical homomorphisms, First Isomorphism Theorem. | section 16.3 |

35 | 30/11/23 | maximal and prime ideals | section 16.4 |

36 | 01/12/23 | polynomial rings, the division algorithm | sections 17.1, 17.2 |

37 | 05/12/23 | the division algorithm. Assignment #6 due. | section 17.2 |

For more details on SageMath, consult the online version of the course textbook, or the SageMath website, or the SageCell website. Here is a link to a SageMath tutorial: https://doc.sagemath.org/html/en/tutorial/

Matt Valeriote