Course: Abstract algebra, Math 417, Section E13
Time: MWF 1:00 PM - 1:50 PM.
Location: 245 Everitt Elec & Comp Engr Lab
Instructor: Chris Leininger
Phone: 265-6763
Email: clein (at) illinois.edu
Office: 366 Altgeld Hall
Office hours: Tu 11-12, Th 2-3, Fr 2-3.

Math 417 is an introduction to abstract algebra. The main objects of study are groups, which are abstract mathematical objects that reflect the most basic features of many other mathematical constructions you are likely familiar with. We will also study rings and fields later in the course, other abstract mathematical objects, which can be thought of as groups with additional structure. The goal of the course is to introduce the students to abstract mathematical thinking through the study of these simple, beautiful mathematical constructions, and to explore the relationship to other areas of mathematics. The required text is available online:

Frederick M. Goodman, Algebra: Abstract and Concrete (Edition 2.6), SemiSimple Press Iowa City, IA. (available online free here).

• Weekly homework (20%): This will be assigned at the beginning of the week, covering the material for that week, and will be due the following Monday, Tuesday, or Wednesday (as posted). This will be graded by the grader.
• Quizzes (10%): There will be short quizzes (at most one per week), typically on Friday. This is to check that you know the absolute basics of what we are doing in class. It will typically cover very recent material.
• Two in-class midterms (20% each): These will be on Wednesday, September 30 and Monday, November 16.
• Final exam (30%):
• This will be a 3 hour exam held 8:00 AM -- 11:00 AM, Tuesday, December 15.
• You can check your scores here.

Extra office hours for final exam:
• Tuesday, 12/8, 11am-12pm (as usual),
• Thursday, 12/10, 2pm-4pm (room 145 AH),
• Friday, 12/11, 1pm-3pm (room 145 AH),
• Monday 12/14, 1pm-3pm (room 145 AH).
Review sheets:
Midterm solutions:

Illegible work will be returned, ungraded

Assignments (please report any errors/mistakes to me)
• Homework 1, due 8/31, beginning of class.
*Notes*
• For problem 1.4.2, you should find the matrices, and do "a few" matrix products illustrating the composition of symmetries is a symmetry. If you want to do the entire multiplicaiton table, you can use mathematica (for example), but this isn't necessary.
• Depending on your background, it may be beneficial for you to read over some/all of the appendices (especially A-E).
• Homework 2, due 9/9, beginning of class.
• Homework 3, Changed: now due 9/16, beginning of class.
• Homework 4, due 9/21, beginning of class.
• Homework 5, due 9/28, beginning of class.
• Midterm 1 solutions.
• Homework 6, due 10/12, beginning of class. Note that 2.7.4 has been removed. It will be added to next week's assignment.
• Homework 7, due 10/19, beginning of class.
• Homework 8... Now due WEDNESDAY 10/28, beginning of class.
• Homework 9... Now due Wednesday, 11/4. Rubik's cube notes can be found here.
• Homework 10, due Wednesday, 11/11.
• Review sheet for midterm 2. See bottom of page for office hours update.
• Nonabelian groups of order 20.
• Homework 11, due 12/2, beginning of class.
• Homework 12, due 12/9, beginning of class.
• Here are some notes on field extensions.
Lectures
• Link to videos of lectures available to students registered for the class.
• 8/23, lecture 1: Examples of groups.
• 8/25, lecture 2: More examples of groups.
• 8/27, lecture 3: Permutation groups.
• 8/31, lecture 4: Primes and divisibility in the integers. (typo page near bottom of page 1: should say "with equality iff a=1 or b=1."
• 9/2, lecture 5: More on primes and divisibility.
• 9/4, lecture 6: Modular arithmetic.
• 9/9, lecture 7: More modular arithmetic.
• 9/11, lecture 8: Basic properties of groups.
• 9/14, lecture 9: Subgroups, isomorphisms, and Cayley's Theorem.
• 9/16, lecture 10: Cyclic groups.
• 9/18, lecture 11: Subgroups of cyclic groups. Typo: On the last page, the bottom subgroup should be the trivial group {[0]} (not [1]).
• 9/21, lecture 12: Dihedral groups. Homomorphisms: examples.
• 9/23, lecture 13: Homomorphisms, kernels, and their basic properties.
• 9/25, lecture 14: Cosets and Lagrange's Theorem.
• 9/28, lecture 15: More cosests and Lagrange's Theorem.
• 10/2, lecture 16: Equivalence relations and partitions. Be ready for quiz on this Monday!
• 10/5, lecture 17: More on equivalence relations and partions.
• 10/7, lecture 18: Quotient groups and homomorphisms.
• 10/9, lecture 19: More examples of quotient groups.
• 10/12, lecture 20: More quotient group theorems.
• 10/14, lecture 21: Products, diamond isomorphism, and direct products of groups.
• 10/16, lecture 22: Direct product again.
• 10/19, lecture 23: Semidirect product.
• 10/21, finished previous notes on semi-direct product.
• 10/23, lecture 24: Recognizing semidirect products. Group actions.
• 10/26, lecture 25: Basics of group actions; examples.
• 10/28, lecture 26: More examples of actions. Orbit-stabilizer Theorem.
• 10/30, lecture 27: More orbit-stabilizer and Burnside/Cauchy-Frobenius theorem. Be ready for a quiz on the orbit-stabilizer theorem!
• 11/2, lecture 28: Proof of Cauchy-Burnside and example. Rubiks cube group defined.
• 11/4, lecture 29: Rubik's cube group.
• 11/6, lecture 30: Class equation and applications to structure of groups.
• 11/9, lecture 31: Sylow theorems and applications to structure of groups.
• 11/11, lecture 32: Proofs of Sylow theorems.
• 11/13, finished proof of Sylow theorems and review.
• 11/18, lecture 33: Rings and fields: introduction and examples.
• 11/20, lecture 34: Polynomial rings over fields: decomposition into irreducibles.
• 11/30, lecture 35: Ring homomorphisms and multiliplicative groups of finite fields.
• 12/2, lecture 36: Ideals and principal ideals.
• 12/4, lecture 37: Quotient rings.
• 12/7, leture 38: Field extensions and fields of fractions.
Other things
• For a quick refresher on proof by contradiction", you can look here.
• I am away 8/31-9/5, and in particular office hours are cancelled this week. You can email me with your questions, and I will have additional office hours on 9/7 2:00pm - 3:30pm and 9/8 10:00 - 12:00.
• Please note that I have moved the first midterm back a couple days. It will now be held on Wednesday, September 30. This will give you time to ask questions about the previous week's material before taking the exam.
• Office hours Thursday, September 17, have been changed. You can come by my office 12:00--12:50 and 2:00--2:30.
• Midterm 1 information:
• Covers sections 1.1 - 1.7, (a little of 1.9), 1.10, 2.1 - 2.5 of Goodman.
• Topics: Definitions and examples of groups (including all the examples we discussed in class), order of a group. Permutation groups: representing elements of $$S_n$$ in disjoint cycle notation, composition, parity (or sign) of a permutation. Primes and divisibility: divisors, prime factorization, gcd. Congruence modulo $$n$$ and arithmetic: the groups $$(\mathbb Z_n,+)$$ and $$(\mathbb Z_n^\times,\cdot)$$ (Goodman writes $$\mathbb Z_n^\times = \Phi(n)$$). Subgroups, cyclic groups, order of an element, dihedral groups. Homomorphisms, kernel, normal subgroups. Subgroups and cosets, index of a subgroup, Lagrange's Theorem.
• The exam will require you to do some calculations (with justifications) and do some short proofs (think "prove Blah is a group" or "prove this easy property").
• Office hours for the rest of this week, and for next week:
• Week 9/21--9/25: I will be unavailable on Friday afternoon, so instead you can come to my office 12:00--12:50 or 2:00--2:50 Thursday (9/24) or 9:00--10:00 Friday (9/25).
• Week 9/28--10/2: Since we have an exam on Wednesday, there will be office hours 2:00pm--4:00pm Monday 9/28, and 10:30--12:00 on Tuesday 9/29. Office hours for the rest of the week will be cancelled.
• Midterm 1 review sheet.
• As I mentioned in class, some people have been confused about what WLOG means. Here are some examples (to avoid):
• Let G be a group with even order. WLOG assume G has order 2. [This does not make sense -- a group with even order need not be a group of order 2, and it's not clear how to go from the case of order 2 to the general case of even order.]
• Let a,b elements of the group G. WLOG let b=a^{-1} [What they want to say is simply “consider the elements a and a^{-1}, but in some sense they don’t feel comfortable with making a ‘choice’ and feel the need to justify its generality"]
• Let a,b elements of the group G. WLOG let c=ab [i.e. “Let c be the product of a and b”. Same observation as above]
• Let a,b elements of order 2 of the abelian group G. WLOG let G={e,a,b} [This does not make sense either since there will likely be many other elements and again it is not clear how to get to the general case from the "special" case (which in fact can never happen)]
• Here is an example of computing the subgroup lattice for Z30
• Here is a picture to help visualize what Propositions 2.7.13 and 2.7.14 tell you.
• Office hours on Thursday have been moved and are now 9-10, instead of 2-3. Sorry for any problems this causes.
• Here is a link explaining step-by-step how to solve the rubik's cube.
• (Extra) office hours for week 11/9-11/13: Tuesday 11-12, Wednesday 12-1, Thursday 2-4, Friday 2-3.
• Office hours for week 11/16-11/20: Cancelled.
• Here is an explanation of how to shwo that two semidirect products are isomorphic.

Back to Prof. Leininger's homepage