Are things getting tougher?

I have now been working through GTB1 of M208 for the last three weeks and I still haven't finished it. I am beginning to wonder if the last two blocks of this course are tougher than the rest. I know that Christmas has slowed things down a bit, but usually I can complete a book in three weeks. One question was a corker. In Exercise 4.7 on p44, you are asked to find all the normal symmetry subgroups of regular hexagon. It sounds innocuous enough, but I ended up covering four sides of A4 with my answer.

GTB1 starts with some much needed revision of Group Theory. It is amazing what you forget when you have head full of analysis. Still, there are some things that are beginning to stick in my leaky brain.

The rest of the chapter starts to delve more deeply into conjugacy and normal subgroups. One essential idea is that conjugacy in symmetry groups represents symmetries that have a similar type. For example, when considering the square, the two reflectional symmetries which are associated with the lines of symmetry that pass through the corners of the square are of the same geometric type and are related by conjugacy. This is built up into the Fixed Point Theorem later in the chapter.

Another major section of the chapter is an exploration of the relationship between conjugacy and normal subgroups and this leads to four properties of subgroups that characterise normality. In the final section there is a look at infinite groups of 2x2 matrices.

I must say that it is tricky stuff. I find it hard to get all these abstract ideas ordered someway in my head so that they can be remembered. There is layer upon layer of ideas and whilst it is ok answering questions when you have just completed a topic, I can imagine that in an exam, it may be hard to come up with the right techniques for answering a random question.

Still, I am very glad that I have done so much advance work on this course as I will be able to spend time mastering the methods when the course actually starts. I am very much looking forward to my first assessment.