## Tuesday, 8 November 2011

### Review of M208 Part 1 Introduction and Group Theory

Ok as promised here is my review of M208. My friend and M208 colleague Neil has already posted his review

http://neilanderson.freehostia.com/courses/maths/m208/

So mine will be complimentary to his. In this post I will review the group theory and Introductory sessions

M208 Covers 3 main areas of pure maths with an introductory section. The areas are Group theory, Linear Algebra and Analysis. It  covers the pure maths element of a typical first year undergraduate degree in maths. The Applied stuff is covered by MST209. This may surprise some people as these are designated second level courses but It's one of the things one has to get used to with the OU that their open door policy means that a lot of catching up has to be done but realisitically MST121 and MS221 are really only up to A level. So in that sense M208 will be the first course beyond A level Maths that one encounters for many people and for me it was certainly the first time I had encountered the Pure maths topics especially analysis.

My motivation for doing this course was to finally understand the language of pure maths especially analysis despite having an MSc in theoretical physics which involved quite a lot of linear algebra and group theory analysis had always been a blind spot and like most practising physicists and Engineers when confronted with a proof involving say the epsilon delta definition of continuity I would either ignore it or scratch my head in puzzlement as to what they were getting at. So my main motivation was to understand real analysis as a prelude to other topics in analysis such as topology. Linear Analysis, Lebesgue Integration and Functional Analysis and in that I believe I succeeded.

Anyway here is a review of the topics

Introduction

This covers the basics of functions, a systematic account of curve sketching. An overview of mathematical language and proof including proof by induction and a small amount of logic, finally an introduction to the number system including an introduction to complex numbers. Most of this has been covered in the last part of MS221 so those who have done this course should find this a bit straightforward.
The curve sketching is treated in a systematic way by the use of sign tables essentially breaking up a function into it's different parts and working out where a part is negative or positive. Whilst tedious this is a really good method for sketching a curve.

Group Theory

This was divided into two parts and for convenience I'll cover both parts in this paragraph. The first part is quite straightforward introducing the concept of a group, a subgroup left and right cosets and Lagranges theorem which is that the order of a subgroup must be a factor of the group as a whole. There is a lot of tedious stuff about the symmetries of planar figures and to be quite frank I would have appreciated it, had they taken a slightly more abstract approach introducing the concept of a dihedral group first of all and then showing how the symmetry operations of rotation and reflection can be related to each other in general. But no we were expected to be able to work out  what happens when a planar figure is rotated through a given angle then rotated about a given axis. Unless you are adept at something like rubiks cube this is a real pain. Yet as I explained in a previous post the rules are really quite simple.

http://chrisfmathsphysicsmusic.blogspot.com/2011/03/progress-or-lack-of-it.html

The other topic covered in the first part was permutations again these can be a bit tricky but practice makes perfect. Also a small amount on matrix groups.

The second part builds on the first part developing the idea of conjugacy classes and extending symmetries to three dimensions. Here again this was a real pain, you now have to think in three dimensions and it took me ages to do a particular TMA question. There is in fact a systematic method for working out the symmetry group of a solid regularly used by Chemists and it would have been much better if the course introduced that rather than leaving everything to chance. Having had a chance to look at the handbook of M336 the next group theory course it seems to cover that there.

The second unit in the last part was probably the most abstract of the group theory units. Introducing the concept of homomorphisms and the relationship between the Kernel of a group and its image set. I need to go back and look at this again, as there was some pretty deep stuff there.

It then ended with the counting theorem again I need to look at this again I gave my rather strong views on the examples they chose namely colouring flags or rotating chessboards which I thought were rather infantile
here

http://chrisfmathsphysicsmusic.blogspot.com/2011/07/m208-tmao5-away.html

I also got my fingers burnt on the M208 forum for pointing this out as some people found it easy.

So overall a mixed unit and probably the least enjoyable from my point of view it seems to me that the underlying structure of group theory is obscured by all the examples they choose and for people like me who find it difficult to visualise 3 d rotations or reflections I felt I wasted a substantial amount of time trying to work out what happens in such cases. Time which would have been better put to use by concentrating on the abstract structure of the subject.

The TMA questions were on the whole applications of concepts rather than conceptual ones testing your ablility to understand the concepts and your ability to understand proofs. Indeed that seems to be the general approach of M208 so although I can apply group theory to problems to say working out the cosets of a group and classifying it's subgroups, I'm still not that confident of my ability to generate proofs of theorems in group theory, for that I'll need a more abstract approach hopefully M336 will help me develop  a better feel for the abstract aspects of group theory.