More on Generating Subgroups

I seem to be blogging more about the groups course M336 rather than the number theory and mathematical logic course M381, but I get the feeling this will all change at some point as I get further into mathematical logic.

Ok, time for an update. I finished the first mathematical logic unit of M381 without too much difficulty. I think the hardest part is dealing with a whole load of new concepts. Once you get the hang of these then it seems to be alright. I then moved onto unit 4 of groups and geometry and this was quite a time consuming book both from the length of it and the considerable number proofs that you have to wade through and also come up with yourself. A lot of the material was revision going over topics such as cosets, normal subgroups, quotient groups, isomorphisms and homomorphisms. However, there was quite a lot of new material and perhaps the most important was proving that you can generate a subgroup of a group using elements of that group (see my previous post).

The approach to this theory was different from what I expected and quite subtle. Firstly, the subgroup of a group G generated by an underlying set of elements S is defined to be the 'smallest' subgroup of G containing the set S. This is denoted <S>. The first step is to prove that there is a unique smallest subgroup of G that contains S. The trick is to consider all the subgroups of G that contain S. This collection is certainly not empty since it contains G (as G is a subgroup of itself). Now consider the intersection H of all these subgroups that contain S. The intersection of a set of subgroups is itself a subgroup (and this was proved in the unit). So H is a subgroup of G. It also contains S because our collection of subgroups each contained S. Further, H must be the 'smallest' subgroup of G containing S because H is the intersection of our set of subgroups and so is contained by those subgroups. Hence <S>=H.

Although this tells us that <S> exists it does not tell us what <S> looks like! The subgroup axioms come in handy here. <S> must contain the identity of G, e say, and the inverse of the elements of S in G. This means that <S> must also be generated by these elements (namely e, the elements of S and the inverse of the elements of S). We can call this set S'. The text then goes on to show that <S> is generated by the set of words in S', W(S'). As words in English are created by taking any finite arrangement of letters (in principle) drawn from our 26 letter alphabet, so the words in W(S') are formed by taking any finite arrangement of elements from S' (but in English we might suspect that eeeeeaaaaeeeooooaeaor isn't a word as it is in group theory!). It turns out that <S>=W(S').

Having been writing this blog over the last few days I have got further into unit 2 of mathematical logic and, surprise, surprise, I have found some of it increasingly difficult (if not impossible)! However, having said that I can't help thinking it is blooming neat and I am glad I am doing it as I think it has resonances in some of the things I have thought about number theory.