MAST20022: Group Theory and Linear Algebra

3 written assignments due at regular intervals during semester contributed to total 20%, a 3-hour written examination (80%).

There were three assignments in this subjects due in week 5, 9, and 12 this year. None of them was particular difficult (the only hard question was made unassessed by Alex at the end), but it still took some time to think about and optimize. Full 20 marks is definitely achievable if you put some effort in them.

As the prerequisite for third year subject Algebra, GTLA is indeed the more "pure math" one among all these second year subjects offered by math department in semester 2. The contents are no doubt quite intelligently simulating, and even more interesting, in fact I’d say that this is by far the most interesting math subject I ever took in uni.

It also has one of the smallest cohorts in all second year math subjects (if not the smallest), mostly taken by students intended to major in mathematics, physics or computer science.

The final exam is divided into two sections, each worth 50 marks. The first section consists of ten short questions which are mostly tutorial questions as mentioned above, these should be 50 marks in the bag if you studied those questions well during semester. The second section consists of four longer questions, each corresponding to one of the four main parts in the lecture notes (except modular arithmetic). These are the harder ones and spilt people who got H1 and those who didn’t. I finished the first section in one hour and gave the remaining two hours to the four questions in second section, with around half hour each which I believe is more than sufficient. So I think it’s appropriate to leave at least twenty minutes for each question in second section. This year’s exam (both the original one and the fire-alarm-induced resitting one) was at a comfortable difficulty level with around 10-15 marks worth trickier parts, similar to the 2014 exam. The one from 2015 is however much harder and requires a lot more effort.


Overall I think GTLA is a wonderful subject to take in semester 2 even if you do not intend to go along the pure math path (if you do then it’s a must). The contents are fascinating and the result will be rewarding if you put in some hard work.

Yes, with screen capture. In this year Alex wrote on the white board in all Tuesday lectures so the recordings only have audio. He mentioned that he originally planned to write on board instead of using document camera in every lecture, but the board in Baldwin Spencer theatre where the Wednesday/Thursday’s lecture held did not allow him to do so.

Dr Alexandru Ghitza

As its title stated, this subject can be mainly divided into two topics: group theory and linear algebra. This year Alex decided to split each of them into two subtopics and teach them alternately, last year from the notes it seemed that he taught all the linear algebra parts altogether at first, then moved to groups.

We first started off from a small topic called modular arithmetic, we learnt about greatest common divisor (gcd), division algorithm, Euclidean algorithm and their applications, then we moved to modular congruences and integers modulo m, the latter will be introduced more throughly in the group topic. At last Alex gave us a glimpse of fields and what means algebraically closed field.

Then started the first part of the linear algebra topic, at first Alex revised some first year linear algebra topics that are relevant to this subject, such as linear transformation, matrix representation and change of basis, but it’s rather quickly (all in one lecture iirc), so I suggest revisit your first year notes if those parts have already become unfamiliar. After this we learnt about eigenspaces, complements of a subspace, direct sum of subspaces, characteristic & minimal polynomial, Cayley-Hamilton theorem and upper triangular form, all these small topics are related to each other and provided motivation & foundation for the key topic of this part: Jordan normal form. This part is very well-coordinated, I was feeling a bit like walking in the fog the first week into it, but after a few lectures the structure and connections in-between all topics became more and more clear and all things made perfect sense. There are two interludes in this part, special relativity, which is fundamentally a linear transformation and solving the simple epidemic stochastic model, which is an application of Jordan normal form. Alex had interludes structured in all the parts after modular arithmetic and are quite enthusiastic in explaining them in depth, it’s actually really nice to know some practical examples of these rather theoretical theorems & lemmas.

In the next part we would get in touch with the concept of group for the first time, Alex started with definitions and some basic examples, after that we learnt about permutation and symmetric groups, then subgroup that are generated by a set and order of elements in a group/subgroup. At this point it starts to get a lot more abstract so please make sure you don’t get behind and grip a firm understanding on each concept. Then we reached another important topic, group homomorphism and isomorphism, which will be used a lot later. After this Alex introduced direct product of groups, cosets of subgroup in a group, the set of cosets which is called quotient, as well as quotient as groups. We would study about normal subgroups, kernel/image of a group homomorphism, their relationship with the two groups in this homomorphic and the first isomorphic theorem for groups. Then we arrived at a relatively new topic about free group and group representations, it may seems bizarre the first time you hear about it, but not completely incomprehensible if one spends some time looking into it. Next comes the final topic of this part, conditions on orders of elements and subgroups, with a few theorems given at first, we would reach a popular application of it: public key cryptography aka RSA. There are quite a lot definitions in this part and most of them are totally new to most of us, so I think it’s a good idea to stay on top of everything after each new lecture and continuingly do revisions on previous ones.

Then we went back to the second linear algebra part and learnt about inner product spaces. In first year we dealt with inner product in real inner product space, now we would also do so in complex ones. After introducing definition and properties, Alex elaborated on orthogonal complements, adjoint transformation and how a linear transformation can be defined as self-adjoint/isometry/normal (these are the generic names, there are also different names over real and complex inner product spaces, but Alex said we would stick to the generic ones in this subjects). Then we moved to the relationship of f-invariant subspace and their orthogonal complement which is f adjoint-invariant, where f is a linear transformation, this led to normal form of isometries on real spaces and the orthonormal basis of these spaces. The proof of this one is a bit complicated and takes some time to get a hold with. At last we would learn about spectral theorem, which is again an extremely important topic in this subjects and very convenient to use in many proofs. After this are two optional topics, which Alex just mentioned briefly this semester due to not enough time left.

Now comes the last part of the subject, which is about actions of groups on sets. Alex talked about the concept of a group G acting on a set X (called G-action on X), the orbit and stabiliser of element x in X, which then led to the orbit-stabailiser theorem. Then we studied counting via group actions, one of the most common applications of it is the number of ways to colour the edges of a regular n-gon by finite number of colours. Then we would look through two particular actions, left multiplication action and conjugation action, and briefly examine the existence of elements of prime order. After these Alex taught us the last (examinable) topic of this subjects, Sylow theorem and its application on group of different orders. There’s also another optional topic at the end which we didn’t get time to cover either.

Alex is, by all means, a wonderful lecturer. He explained everything (even those seemingly make nonsense at the first time) really well in lectures, and was extraordinarily helpful during consultation hours. No matter how stupid the question you asked, he would always try to understand your standing and give an appropriate explanation. Also I love his "blackboard & chalks in office" idea, just simply brilliant. Not only could he demonstrate and work on the board, but also students who wanted to give their ideas a go on site. Sometimes if there were a few students in the consultation at the same time, he would let them to ask in turn, I found this is actually a very good way to learn and make me look into the contents in different angles. This semester in GTLA it’s probably the most consultation hours I’ve attended in my entire life, it’s far more enlightening than I could ever describe.

Yes, 2 past exams from 14 and 15, both with solutions.

The tutorial runs once a week, which is your typical math subject tute: 3-4 students each table are given tutorial sheet for the week and work these questions on the white board together. The tutorial questions followed closely to contents in previous week’s lectures with maybe one or two tricky ones marked with stars. Alex would post both the questions and the answers on subject’s homepage after each week, so no answers were given after the tutorials. Our tutor Matthew, tends to do some brief revisions of last week’s contents on the board first then let us do the tutorial questions. If some questions/points confused a few students, he would demonstrate them on the board to the whole room. Each tutorial sheet contains a fair amount of questions and it’s almost impossible to finish them all in a one-hour tutorial. But it’s crucial to try and finish them in your own time and understand all of them clearly, since half of the final exam are questions taken directly from tutorial sheets or modified only very slightly.

5 Out of 5

Lecture notes are updated throughout the semester and can be accessed on the subject homepage. The subject homepage also listed other references/resources as well as the lecture notes from previous year.

3 x one hour lectures per week, 1 x one hour practice class per week.

2016, Semester 2

H1 (86)

Three written assignments (totalling 20%); One 3 hour examination (80%)

This subject is the first pure maths subject that most students study after the shock of either Accelerated Mathematics 2 or Real Analysis, but most students tended to find that the level of mathematical rigour was considerably lower than in those subjects. As the title suggests, the subject covers advanced linear algebra as well as an introduction to abstract algebra (in this case, fields and then groups). Both Linear Algebra and Group Theory are very important in physics as well as in pure mathematics, and are quite fascinating fields of study.

Topics covered:

• Fields: Elementary number theory (division, prime numbers, modular arithmetic)
• Fields: What is a field?
• Linear Algebra: Review
• Linear Algebra: Linear transformations and the Jordan Normal Form
• Linear Algebra: Complex inner product spaces and the Spectral Theorem
• Group Theory: What is a group?
• Groups: Permutation groups, matrix groups, cyclic groups
• Groups: Direct products Cosets, Lagrange's Theorem with applications, RSA Cryptography, isomorphisms
• Groups: Normal subgroups, quotient groups
• Groups: Group actions, symmetry groups, Orbits & Stabilisers, Groups actions on groups w/ applications
• Groups: Groups of Euclidean isometries

Yes, this sounds like a lot of terminology which doesn't exactly sound very exciting, but after a while connections appear between the disparate topics studied and the subject becomes all the better for it. Do be careful, though, as the subject is initially quite easy and slowly increases in difficulty until the later topics in Group Theory are difficult to grasp. The final week, covering Euclidean geometry, is pretty boring and completely useless wrt the exam, but the rest of the course is certainly assessed. The assignments also increased in difficulty as the semester progressed, but were largely manageable (plus we got to make a cube in the 3rd assignment!). The tutorials were of course very useful and certainly my tutor was excellent. My main criticism, though, is of course the pacing of the subject - too much material was left until late in the semester, and at the half way point of the course we hadn't even started Group Theory. This was one of the few weak points of a great subject.

No. See the Textbook Recommendation for details.

A/Prof Craig Hodgson. My main issue with the lecturer was that his monotonous voice made for a very soporific atmosphere which discouraged me from attending most of the lectures. Watch out for the ponytail...

Yes, there were exams from 1999 onwards. In fact the exams from 1997 - 1999 (with answers) were provided in the course notes as well.

5 Out of 5

None. The course notes were available for purchase from the Co-op Bookshop as well as in PDF form on the LMS. While the lectures weren't recorded, everything the lecturer wrote on the blackboard was scanned and posted on the LMS immediately afterwards, so with these two resources it really wasn't necessary to attend lectures. Indeed I only attended 6 out of 36...

3 x 1 hour lectures per week; 1 x 1 hour tutorial per week

Semester 2, 2013

85