Vote here for in-class presentations

Vote here for presenter 1 on Mon, Dec 3, 2012: http://www.surveymonkey.com/s/279HS6Q
Vote here for presenter 2 on Mon, Dec 3, 2012: http://www.surveymonkey.com/s/27RSKDX
Vote here for presenter 3 on Mon, Dec 3, 2012: http://www.surveymonkey.com/s/273DKWG
Vote here for presenter 4 on Mon, Dec 3, 2012: http://www.surveymonkey.com/s/276K92T
Vote here for presenter 4 on Mon, Dec 3, 2012: http://www.surveymonkey.com/s/29JMCFZ

In-class presentations & Rubik's cube comp

5% of your total grade will be based on a 5-10 minute in-class presentation.

The contents of the presentation could be taken from the textbook (e.g. a topic/concept/exercise I have not discussed in class), or from anywhere else, as long as it is on or related to abstract algebra.

To encourage you to start doing these presentations before we run out of class time, we will have the following scheme for bonus points:

The n-th person to email me to put their name down to present will receive an extra 2-(n/10) % to the final grade.  (So the 1st person to let me know will receive an extra 1.9%, the second person 1.8%, and so on.  However, if you let me know you'll present on a certain date and that does not materialise, I'll move you to the back of the queue.)

Here is a list of the remaining days of class, and the number of people who have signed up to do presentations on any particular day:

• Fri, Nov 2
• Mon, Nov 5 (1 person)
• Wed, Nov 7 (1 person)
• Fri, Nov 9
• Mon, Nov 12 (2 people)
• Wed Nov 14 (1 person)
• Fri, Nov 16 (2 people) (last day of class before Thanksgiving)
• Mon, Nov 26 (2 people)
• Wed, Nov 28 (tentative date for Rubik's cube comp*; we have an MC)
• Fri, Nov 30 (4 people)
• Mon, Dec 3 (5 people)
• Wed, Dec 5 (Exam 3)

*As of Fri, Nov 9, we seem to have 4 entrants for the competition.


Rubik's cube comp
As mentioned on the syllabus and in the post `Course Information', the winner (1st place) of the competition will earn an extra 5% towards the final grade.  The runner-up (2nd place) will earn 3%, and the second runner-up (3rd place) will earn 2%.

Assignments

The above should be `Assignment 3'!

Lecture log & Exercises

Note: the exercises listed below are not to be submitted for grading.  However, you are very welcome to talk to me about them, and you should certainly work through all of them.

• Lec 1: Mon, 20 Aug 2012: principle of induction, equivalence relation
• Lec 2: Wed, 22 Aug 2012: examples of equivalence relations, equivalence class, in-class quiz 0.1
• Lec 3: Fri, 24 Aug 2012: partition, example of rational numbers as equivalence classes, divisibility
• exercises: induction (Sec 1.1: 2, 4); equivalence relations: (Sec 1.1: 7, 9)
• Lec 4: Mon, 27 Aug 2012: greatest common divisor (gcd), example of Euclid's algorithm
• Example 1(iv) on p.16 can be done with:  http://www.algebra.com/algebra/homework/divisibility/gcd-euclidean-algorithm.solver
• Lec 5: Wed, 29 Aug 2012: Euclid's algorithm, Euclid's lemma
• exercises: Sec 1.3: 1(i)(iv), 3, 5, 7(i)(ii)
• Lec 6: Fri, 31 Aug 2012: least common multiple (lcm), prime numbers, fundamental theorem of arithmetic
• exercises: Sec 1.4: 5, 11
• Lec 7: Wed, 5 Sep 2012: Euler's phi function, Euler's Theorem, Fermat's Little Theorem
• exercises: Sec 1.7: 1, 4, 7
• Lec 8: Fri, 7 Sep 2012: public key cryptography (RSA encryption)
• Lec 9: Mon, 10 Sep 2012: definition and basic properties of congruence
• exercises: Sec 2.1: 2, 3, 5, 6
• Lec 10: Wed, 12 Sep 2012: congruence being an equivalence relation, congruence classes
• Lec 11: Fri, 14 Sep 2012:  principal representative of a congruence class solving linear congruence equations
• exercises: Sec 2.1: 9(iii), (iv), (v)
• Lec 12: Mon, 17 Sep 2012: techniques for solving linear congruences, multiplicative inverse, using Euclid's algorithm to find a multiplicative inverse
• exercises: Sec 2.1: 11(ii), 13(i), 15(i)(ii)
• Exam 1: Wed, 19 Sep 2012.
• Lec 13: Fri, 21 Sep 2012: Chinese remainder theorem, applications of modular arithmetic (bar codes, checking divisibility), definition of \mathbb{Z}_m and addition/multiplication in it.
• exercises: Sec 2.1: 17, 18
• Lec 14: Mon, 24 Sep 2012: properties of addition and multiplilcation in \mathbb{Z}_m, addition and multiplication tables of \mathbb{Z}_5
• exercises: Sec 2.4: 3(i),(iii), 5(ii),(vi)
• Lec 15: Wed, 26 Sep 2012: units and zero divisors in \mathbb{Z}_m, number of units in \mathbb{Z}_m is \varphi (m), solving linear equations in \mathbb{Z}_m, axioms of a ring
• exercises: Sec 2.4: 6, 9
• Lec 16: Fri, 28 Sep 2012: examples of rings (including \mathbb{Z}_m, M_{2 \times 2}(\mathbb{R}) and the path algebra of a quiver)
• Lec 17: Mon, 1 Oct 2012: the path algebra of a quiver (continued), zero divisors
• Lec 18: Wed, 3 Oct 2012: integral domains, fields
• exercises: Sec 2.5: 5, 6, 7, 12
• Lec 19: Fri, 5 Oct 2012: axioms of a group, examples of groups and their operation tables, abelian groups
• Lec 20: Mon, 8 Oct 2012: more examples of groups (of 2 by 2 and 3 by 3 matrices), \mathbb{U}_m (the multiplicative group of units in \mathbb{Z}_m)
• Lec 21: Wed, 10 Oct 2012: the Klein four group (the symmetry group of a rectangle that is not a square), and how it is isomorphic to \mathbb{Z}_2 \oplus \mathbb{Z}_2
• exercises: Sec 4.1: 2, 3, 10
• Lec 22: Fri, 12 Oct 2012: consequences of group axioms (Proposition 1, p.192), the dihedral group D_3 (the symmetry group of an equilateral triangle)
• exercises: Sec 4.1: 8, 12; Sec 4.2: 1, 2, 4
• Lec 23: Mon, 15 Oct 2012: subgroups, "subgroup criterion", proper/nonproper subgroups, trivial subgroup
• exercises: Sec 4.4: 2, 4
• Lec 24: Wed, 17 Oct 2012: subgroup criterion for finite groups, cyclic subgroups and generators, 
• exercises: Sec 4.4: 5, 12
• Lec 25: Fri, 19 Oct 2012: intersections of subgroups, subgroups generated by a set, order of an element 
• exercises: Sec 4.4: 14, 20
• Lec 26: Mon, 22 Oct 2012: direct products of groups, direct sums
• exercise: Sec 4.4: 26
• Exam 2: Wed, 24 Oct 2012 (BYOT)
• Lec 27: Fri, 26 Oct 2012: more on direct products/sums, operation-preserving functions
• exercises: Sec 4.4: 27
• Lec 28: Mon, 29 Oct 2012: examples and non-examples of operation-preserving functions, isomorphisms
• Lec 29: Wed, 31 Oct 2012: examples of isomorphisms of groups, properties of isomorphisms, classifying finite groups
• exercises: Sec 4.5: 1, 2
• Lec 30: Fri, 2 Nov 2012: homomorphisms of groups and examples
• exercises: Sec 4.5: 11, 12
• Lec 31: Mon, 5 Nov 2012: kernel and image of a group homomorphism, classification of cyclic groups
• Lec 32: Wed, 7 Nov 2012: definition of permutations, definition of permutation group S_n, isomorphism between S_3 and D_3 (the dihedral group that is the symmetry group of an equilateral triangle), different notations for permutations (including cycle notations)
• exercises: Sec 4.7: 1
• Lec 33: Fri, 9 Nov 2012: non-disjoint vs disjoint cycles, cycle decomposition of a permutation, every permutation is a product of transpositions (2-cycles)
• exercises: Sec 4.7: 2, 5, 6
• Lec 34: Mon, 12 Nov 2012: even and odd permutations
• exercises: Sec 4.7: 4, 8
• Lec 35: Wed, 14 Nov 2012: group homomorphism from (S_n,\circ) to (\mathbb{Z}_2,+), the n-th alternating group A_n
• Lec 36: Fri, 16 Nov 2012: Cayley's Theorem, isomorphism between the subgroup ({R_0, R_1, R_2},*) of the dihedral group (D_3,*) and the subgroup ({e,(1 2 3), (1 3 2)},\circ) of (S_3,\circ)
• exercises: Sec 4.7: 12, 13
• Lec 37: Mon, 26 Nov 2012: (in-class presentations and teaching evaluation)
• Lec 38: Wed, 28 Nov 2012: (Rubik's cube competition) Counting colourings of an equilateral triangle (Section 4.8: p.240), equivalent colourings.
• Lec 39: Fri, 30 Nov 2012: (in-class presentations) Task 1 on p.241, counting formula (at the end of p.242) applied to the equilateral triangle 
• Lec 40: Mon, 3 Dec 2012: (in-class presentations) Counting isomers of a benzene ring (Task 12, part 6 on p.245).
• exercises: Sec 4.8: 
• Task 2
• Task 9 (i), (ii)
• Task 10
• Exam 3: Wed, 5 Dec 2012.

Applications

This post will be a collection of articles/resources on the applications of abstract algebra, which would be of general interest.  Feel free to send me links that you think should be included.


Coding theory:

1. (communications) NASA and Mars rover

Number theory:

1. (cryptography) Online banking security
2. (chemistry) balancing chemical equations using Diophantine equations.  

Group theory:

1. (chemistry) Molecules and Rubik's cube
2. hexaflexagon:
• a template
• instructions on folding (This is just one way to fold it? I found the instructions on this page easier to follow than those on some other webpages.)  This page also tells you how to flex a hexaflexagon.
• a YouTube video on the flexahexagon
3. braid groups:
• a braid applet by Stephen Bigelow

In-class quizzes

Quiz 0 (test): http://www.surveymonkey.com/s/KSHK3SR
Quiz 0.1 (test): http://www.surveymonkey.com/s/5JSCKHP
Quiz 1: http://www.surveymonkey.com/s/SWXH6Q2
Quiz 2: http://www.surveymonkey.com/s/CKFLYCK
Quiz 3: http://www.surveymonkey.com/s/KHM7MZW

Course Information

Instructor information: Jason (Chieh-Cheng) Lo
Office location: Room 320A, Math Sci Building
Department webpage: www.math.missouri.edu/~locc
Office hours: 11am-noon Mon, and 10am-noon Wed



Course description: This is an introduction to abstract algebra, intended for Mathematics Education majors.  The goal is for the students to learn about the basic algebraic structures such as rings, fields and groups, through a lot of concrete examples, so as to prepare them as secondary school teachers.

Class time/location: 1-1.50pm, Mon/Wed/Fri, at Crowder Hall, Room 101

Textbook (required): An Introduction to Abstract Algebra with Notes to the Future Teacher, by Nicodemi, Sutherland and Towsley.

Prerequisites: Math 2300 or 2320

Assessments and grading: 

• Three in-class exams, weighted equally, with a combined worth of 80% of the total grade.  The dates are:
• Exam 1: Wed, 19 Sep 2012
• Exam 2: Wed, 24 Oct 2012 
• Exam 3: Wed, 5 Dec 2012
• Three assignments or quizzes, each worth 5%
• A 5-10-minute in-class presentation, worth 5%.

Letter grades will be awarded based on the following cut-offs: 90% is the cut-off for an A-, 80% the cut-off for a B-, etc.  These cut-offs may be adjusted if the need arises.

Extra credit: We will also have a Rubik's cube competition.  The details will be announced during the semester. The competition winner will earn further 5% towards the final grade.

In-class quizzes: We will have  in-class quizzes (1-3 times a week), given after we finish each section or topic.  These will not be counted towards the final grade.  Their purpose is for me to keep track of your individual progress.