We're off!

The course hasn't officially started yet, but it feels like it. The course website opened last Tuesday and I was finally able to get my hands on the first TMA. Since then, I feel like things have stepped up a gear and I am already having to devote more hours to my study. For starters, there is TMA01 part 1 to do and it was good revision. I did get myself in a bit of a twist to start with but things worked out ok in the end. It is amazing how one tiny slip can lead you down a blind alley. Then there is the sudden burst of activity on the forums to follow; a bit more revision of I1 to do; talking to my tutor about one of the problems I found in one of the exercises (that was far more interesting than I realised) and trying to keep up with my work with the books at the end of the course. Phew! I won't be able to keep this up at this rate. I may have to scale back to just doing the TMA's and the end of course work.

Other than that I have finished the chapter on Homomorphisms (GTB2) and have begun the chapter on Group Actions (GTB3). This is my final group theory book. I don't mind group theory too much and I find it ok to work with. In the chapter on Homomorphisms (I wish that word wasn't such a mouthful!) we find that they are a bit like Isomorphisms but with the strict one-one and onto condition relaxed. Homomorphisms are basically functions that map one group to another and in the mapping process some of the domain group properties are preserved. For example, the identity in the domain is mapped to the identity in the codomain, as are inverses and powers of elements. Also elements that are conjugate in the domain are also conjugate in the codomain.

The meat of the chapter comes with the discussion of Kernels and Images of a homomorphism. This is a bit like the discussion of Kernels and Images of a linear transformation in the Linear Algebra blocks. The Kernel of a homomorphism is the set of elements of the domain that map to the identity element in the codomain. The Image is the usual notion of an image of a function. It turns out that the Kernel is a normal subgroup of the domain group and the Image is a subgroup of the codomain group. The fact that the Kernel is a normal subgroup of the domain means, of course, that its cosets partition the domain and this leads on to the Correspondence Theorem, a return to Quotient Groups and finally the Isomorphism Theorem (I ain't going to discuss all that!). It's all a bit of a handful to remember. Reasonably straight forward at the time but instantly forgettable!