Group Theory and Linear Algebra
Semester II 2020
Lecturer: Arun Ram, 174 Peter Hall, email: email@example.com
Time and Location:
Lecture: Tuesday 13:00-14:00 Old Geology G04 Theatre 1,
Lecture: Thursday 9:00-10:00 Biosciences 4-G003 Agar Theatre,
Lecture: Friday 14:15-15:15 Old Geology G04 Theatre 1.
Practice class: Monday 11:00-12:00 in David Caro-Podium 203
Practice class: Monday 15:15-16:15 in David Caro-Podium 204
Practice class: Monday 16:15-17:15 in Peter Hall 215
Practice class: Tuesday 14:15-15:15 in David Caro-Podium 203
Practice class: Thursday 13:00-14:00 in Peter Hall 215
Practice class: Thursday 12:00-11:00 in Peter Hall G11
Practice class: Friday 16:15-17:15 in Peter Hall G14
Practice class: One per week, see the
Consultation hours: Arun Ram, ??????, email: firstname.lastname@example.org
Student Representatives: ????????, email: ????@student.unimelb.edu.au
The homework assignments will soon appear below (7% each = 5 +2???)
- Assignment 1: Due ???14 and 21??? August
- Assignment 2: Due ???11 and 18?? September
- Assignment 3: Due ???16 and 23??? October
A combination of factors, one of them being the Corona virus have made me
start to think, early, how I am going to deliver my Group Theory and
Linear algebra course next semester. Another major stimulus was the
fact that Persi Diaconis is planning to teach a similar course starting
at the end of March 2020, and I wanted to offer my notes to him as a resource.
The notes are structured in a form analogous to a dictionary.
They are NOT meant to be read in a purely linear order starting from
page one. The section on modules, does not require the section on
groups, the section on Fields does not require the section on Rings.
Unfortunately, physical parameters means that, for a published
book one has to choose an ordering of the pages, and some people
take this as a directive to read the pages in purely linear order.
But this is an arftifice, and the reader that reads the pages in a linear
order from page 1 will get a much lesser experience,
and a significantly less valuable resource.
Happily, on a web page one can naturally interleave the links in a
more stimulating fashion, making the whole exercise feel like a fun Sudoko
Already, because this includes proofs written in proof machine,
this is totalling to about 160 pages. If we add the exercises
for these topics we have somewhere in the neighborhood of 200 pages.
That is probably enough for a single book.
Other material can be put in different books.
Sets and number systems will contain Osmosis topics
and examples of rings and modules:
- Sets: subsets, unions, intersections, products
- functions: injective, surjective, bijective, images
- equivalence relations, partitions, fibers of functions
- partially ordered sets, Zorn's lemma, supremum, infimum
- ordered fields, inequalities
- The positive integers, the nonnegative integers, the integers,
and ideal in the integers
- The fields ℚ, ℝ and ℂ.
- The p-adic numbers ℤp and ℚp
The polynomial ring
the group algebra of , and a study of its ideals
i.e. the group algebra of ,
and some study of its ideals
- Matrices, direct sums of matrix algebras, and ideals in direct
sums of matrix algebras
Then a Linear algebra
book can focus on orbit problems
for subcategories of vector spaces:
- -modules (vector spaces, bases,
linear transformations, functors: tensor powers, symmetric powers,
exterior powers, duals, adjoints
Automorphisms of vector spaces (row reduction and generators and
relations for ),
-modules (Smith normal form and the
fundamental theorem of abelian groups),
(eigenvectors, Rational canonical and Jordan normal forms),
vector spaces with bilinear form (Gram-Schmidt, Sylvester's theorem)
I had last taught the Group Theory and Linear Algebra course in 2011
and, going back to my web page for
that course, I found an extensive and pleasant list of problems,
and a complete set of lecture notes, handwritten and typed. The
Problem lists were in html, and my old original notes needed conversion
from Plain TeX to LaTeX. I have now done this and am now at a stage where
I only needs to add pictures, clean up some proofs and proofread, proofread,
In my notes,
the parallels between the structure theorems for Groups, Rings, Group actions,
modules, Fields and Vector spaces are very carefully worked and presented.
The examples for group families and groups of low order are well organised,
thoughtful and thorough.
I'm less happy with the fact that there is no good section on
examples for rings, and no good section on examples for modules.
But one step at a time, we shall see if we can put these in in due time.
However, there is a very good argument that adding this material would make
it too long to be a high impact resource for learning (no-one ever really
reads long books).
One standard addition to the course is to put in the Sylow theorems.
For a long time I have felt that the Sylow theorems should be accompanied
by the analogous theorems for Borel subgroups and tori (in algebraic
groups) so that one sees vividly the analogies between $p$-subgroups,
Borel subgroups and tori.
After some thinking about it, I decided that is is too much (both for the
writer and reader) to put Borel subgroups and tori into this introductory
text, and found that the Sylow theorems can very sensibly and motivationally
included as exercises (with maps and directions for the proofs) to the
seciton on Group Actions.
I struggle with where to place the material on sets, functions (morphisms
in the cateogry of sets), cardinality (isomorphism classes in the category
of sets), partially ordered sets, and ordered fields. When I teach
this course, I often treat these topics in a couple of lectures
entitled "Osmosis topics" (topics that you are somehow supposed to learn
by osmosis. However, these will have to be properly treated in a different
book, which be attractively titled Sets and Numbers or
Numbers and Sets (which is better?).
- The notation doesn't match with
the category of sets, where one can use . This is an example of "mixing categories". This notational
mishap gets exacerbated with the notation which means
- The inappropriate definition of a subgroup as a subset
that satisfies: if
then TRY MATH JAX HERE obfuscates the structure and is counterproductive.
- For some time I have felt that rings are unhelpfully named and that
ℤ-algebra would be much more sensible. I've rewritten the rings
section to try to stimulate some shift in terminolgy in this direction.
It is pleasant that both the French words algèbre and anneaux start
with A, making it natural to comfortably denote a ℤ-algebra as A.
- No books, notes, calculators, ipods, ipads, phones, etc at the exam.
- Tips to avoid freaking out:
- Just as a carpenter without a hammer is mostly useless, a mathematician
without a good place to work is mostly useless.
- The assignments are designed to take "an average of 6 hours per week". This is an average.
- The assignments can be reformatted to reduce the freak factor: See Assignment 1 (pdf file) from 2009 Real Analysis and Applications as an example.
- The assignments and the course are designed to make you know exactly what is on the exam, practice what is on the exam and do well on the exam.
- The assignments are worth 20% of the total mark. If you skip a few questions it will affect your total mark very little.
- Thousands of students have made it through this course format with Professor Ram in the past (and are proud to tell the tale). You can do it too.
- Tips for time management:
- It is much easier (and safer) to run 45 min per day to attain 6 hours in a week and 24 hours in 4 weeks, than to run for 24 hours solid every fourth week on Sunday.
- To actually run 45 min, it takes me at least 15 min to psyche myself up and convince myself that it is actually not raining and so therefore I should go running, and after a 45 min run I always walk for 5 min and I always go home and have a glass of milk and tell my wife (at length) how cool I am for running 45 min per day. All in all, I waste a good 40 min when I go running for 45 min. If I were more efficient (and every so often, but rarely, I am) then it would only takes me 50 min.
- Measurement of time is a tricky thing and requires real discipline. Teaching and research faculty at University of Melbourne recently had to complete a survey on distribution of their time on the various activities of the job: Do I count the 6 times I had to go check my email and the weather and my iPhone in the time that I spend preparing my Group Theory and Linear Algebra lecture?
- Tips for exam preparation:
- The time that a 100m olympic runner (who wins a medal) is actually competing at the olympics is say (5 heats, 7sec each) 40 seconds. Successful performance during these 40 sec is impossible without adequate preparation.
- The time that a Group Theory and Linear Algebra student spends on the final exam is 3 hours. Sucessful performance during these 3hours is .....
- Consultation hours for Prof. Ram will be ??????????.
- Prof. Ram reads email but generally does not respond by email. Generally,
an effort will be made to respond to email during class time. Almost always
the question that is asked is a question that many students have and
so it is much more efficient to provide the answer during class time.
- The start of semester pack includes: Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file),
Beyond third year (pdf file), Vacation scholarships (pdf file), SSLC (pdf file).
- It is University Policy that:
“a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results. Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).
Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
The handbook entry for this course is at https://handbook.unimelb.edu.au/2020/subjects/mast20022. The subject overview that one finds there:
This subject introduces the theory of groups, which is at the core of modern algebra, and which has applications in many parts of mathematics, chemistry, computer science and theoretical physics. It also develops the theory of linear algebra, building on material in earlier subjects and providing both a basis for later mathematics studies and an introduction to topics that have important applications in science and technology.
Topics include: modular arithmetic and RSA cryptography; abstract groups, homomorphisms, normal subgroups, quotient groups, group actions, symmetry groups, permutation groups and matrix groups; theory of general vector spaces, inner products, linear transformations, spectral theorem for normal matrices, Jordan normal form.
- (1) Greatest common divisors, Euclid’s algorithm, arithmetic modulo m.
- (2) Definition and examples of fields, equations in fields.
- (3) Vector spaces, bases and dimension, linear transformations.
- (4) Matrices of linear transformations, direct sums, invariant subspaces, minimal polynomials
- (5) Cayley-Hamilton theorem, Jordan normal form.
- (6) Inner products, adjoints.
- (7) Spectral theorem. Definition and examples of groups.
- (8) Subgroups, cyclic groups, orders of groups & elements, products, isomorphisms.
- (9) Lagrange's theorem, cosets, normal subgroups, quotient groups, homomorphisms.
- (10) group actions, orbit-stabilizer relation, conjugation.
- (11) Some results on classification of finite groups, Euclidean isometries
Assessment will be based on
three written assignments due at regular intervals during semester
amounting to a total of up to 50 pages (20%),
and a 3-hour written examination in the examination period (80%).
Almost everybody seems to agree that the "50 pages" doesn't make much sense
in our modern technological world where some assignments might even be submitted
in video form (the handbook entry has not been changed for
more than 10 years) and
a better estimate is to note that the handbook entry says that Total time commitment (should average) 170 hours. If we spread this over 14 weeks (12 class
and 2 weeks exam study), we get 12.14 hours per week, and then if we
subtract the number of class and tutorial hours per week (4) then we get
8.14 hours per week, and if we spread 12 weeks of 8 hours over 3 assignements
then we 4*8.14=32.6, as the average number of hours of study and work
to complete each assignment.
If you are speding more time than this, then
please contact us immediately
so that we can help you to make your study more efficient
and effective so that you
will be able to complete the necessary work for this class in
this expected time commitment timeframe.
The homework assignments will soon appear below:
- Assignment 1: Due 23 August
- Assignment 2: Due 4 October
- Assignment 3: Due 1 November
Resources part I: recommended Texts
The following problems page may have helpful examples:
Resources part II: Lectures and lecture notes from 2011
- Lecture 1, 4 August 2020:
The clock and invertible elememnts
Math Grammar: Definitions, Theorems and How to do Proofs (pdf file) and
handwritten lecture notes-pdf file
Examples of proofs written in proof machine (pdf file)
- Lecture 2, 6 August 2020:
gcd and Euclid's algorithm - handwritten lecture notes - pdf file
- Lecture 3, 7 August 2020:
Equivalence relations - handwritten lecture notes - pdf file
- Lecture 4, 11 August 2020:
Functions - hand written lecture notes - pdf file
- Lecture 5, 13 August 2020:
Rings and Fields (pdf file)
- Lecture 6, 14 August 2020:
C[t], gcd and Euclid's algorithm (pdf file)
- Lecture 7, 18 August 2020:
Vector spaces and linear transformations (pdf file)
- Lecture 8, 20 August 2020:
Span and bases (pdf file)
- Lecture 9, 21 August 2020:
Change of basis (pdf file)
- Lecture 10, 25 August 2020:
Eigenvectors and annihilators (pdf file)
- Lecture 11, 27 August 2020:
Minimal and characteristic polynomials (pdf file).
- Lecture 12, 28 August 2020:
Jordan normal form (pdf file)
- Lecture 13, 1 September 2020:
Block decomposition (pdf file)
- Lecture 14, 3 September 2020:
Cayley-Hamilton theorem (pdf file)
- Lecture 15, 4 September 2020:
Inner products and Gram-Schmidt (pdf file)
- Lecture 16, 8 September 2020:
Orthogonal complements and adjoints (pdf file)
- Lecture 17, 10 September 2020:
The spectral theorem (pdf file)
- Lecture 18, 11 September 2020:
Groups and group homomorphisms (pdf file)
- Lecture 19, 15 September 2020:
The polar decomposition (pdf file)
- Lecture 20, 17 September 2020:
Symmetric groups and subgroups generated by a subset (pdf file)
- Lecture 21, 18 September 2020:
Cyclic groups and products (pdf file)
- Lecture 22, 22 September 2020:
Cosets and quotient groups (pdf file)
- Lecture 23, 24 September 2020:
Quotient groups (pdf file)
- Lecture 24, 25 September 2020:
- Lecture 25, 29 September 2020:
Group actions, orbits, stabilizers (pdf file)
- Lecture 26, 1 October 2020:
Centres and p-groups (pdf file)
- Lecture 27, 2 October 2020:
Proof of the Orbit-Stabilizer theorem (pdf file)
- Lecture 28, 13 October 2020:
The affine orthogonal group and isometries (pdf file)
- Lecture 29, 15 October 2020:
Isometries of E2 (pdf file)
- Lecture 30, 16 October 2020:
Matching the affine orthogonal group with isometries (pdf file)
- Lecture 31, 20 October 2020:
Revision: Analogies (pdf file)
- Lecture 32, 22 October 2020:
Revision: The Fundamental Theorem of Algebra (pdf file)
- Lecture 33, 23 October 2020:
Revision: Proof machine (pdf file)
- Lecture 34, 27 October 2020: Revision: Working sample randomly chosen problems
- Lecture 35, 29 October 2020: Revision: Maths, Music and the Weil conjectures
- Lecture 36, 30 October 2020:
Revision: C[t]-modules (pdf file)
Resources part III: Other notes
Various lecture notes from the past that will be useful and supplemented during the term.
Every subject at the University of Melbourne uses a student
questionnnaire to let teaching staff know what students think about the
quality of teaching in that subject. This is now administered online near the end of the semster. As such, it is too late to affect the
teaching for the cohort of students that answer the questionnaire.
Feedback to students based on 2009 questionnaires for Real Analysis:
- The student survey last year showed high student satisfaction with the
course. Most elements of last year's course are being retained.
- Exam performance demonstrated that students had learned concepts and the general framework well, but were weak on skill (they knew what a hammer is for but were unable to use it to hammer in a nail effectively). Skill level is an important goal for this course and this semester there will be a determined effort to get the skill level of all students to a high level:
- The problem sheets will be very directed towards the final exam.