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The essential thing is that what interests us is the group as a whole, and not the individual vectors that make up the group. Somehow or other the group as a whole has a shape for want of a better wordand this shape is what we care about.
The vectors in the group will always all have the same dimension, and in most cases this dimension will be what's called the rank of the group, which can be very large. The are four general rules that must be respected: This applies to integer multiples, not necessarily to fractional multiples.
To be interesting, a group should not be finitely generated. Groups where the entries in the vectors are all integers also turn out to be extremely simple, and thus uninteresting. As illustrated above, in constructing examples, it is usually the denominators which appear in various positions in the vectors which determine the overall shape of the group.
Finite rank torsion free groups are thus very simple mathematical structures, but the resulting theory is not at all simple. The fundamental question in the theory is to describe all possible finite rank torsion free groups in a coherent structured way, i.
Since this question is much too difficult to ever be answered, it can be reframed as the problem of finding qualitative ways to describe groups which will provide useful information about the group. A starting point is the observation that two groups which are not the same can nonetheless be essentially identical in their algebraic structure.
In this case, the groups are called isomorphic. Many of the questions which one asks about finite rank torsion free groups will include the phrase "up to isomorphism. This means that G is the direct sum of two subgroups H and K, where H consists of all elements whose last two coordinates are zero, i.
And K consists of all elements whose first coordinate is zero, i. On the other hand, K cannot be broken up into a direct sum. This is not completely obvious, but it's not difficult. It follows that the first piece must contain the element 0,1,0 and the second piece must contain 0,0,1. But then one gets a contradiction to the requirement that each element in K can be obtain in only one way as a sum of elements from L and M.
Not many readers will be able to follow the preceding paragraph, especially since I've been rather sloppy in the way I've stated things. But what I'm giving your here is not a tutorial on finite rank torsion free groups. I simply want to give you some vague sense of a few of the basic words. The most important point though is that direct sum decompositions are usually not as obvious as the ones in this example.
It is reasonable to ask why anyone would want to study groups of vectors of this sort. And I don't have a really good answer, except that it is a long-standing subject of interest in mathematics.
My own motivation was also simply that the subject was interesting and very challenging and has connections to a large number of other parts of algebra.
The example I've given may seem simplistic, and in fact it is far simpler than most examples in the theory.
But in any case, many examples in other parts of mathematics, such as the theory of finite non-abelian groups, which has very important applications in physics and other sciences, look equally simple if presented in a way stripped down to their bare bones.
And the methods which one develops in the process of studying finite rank torsion free groups are not necessarily limited to this area. A Generation Gap An interesting sidelight here is that fact that I came into mathematics about roughly the time of a great generational gap among mathematicians: My graduate work at the University of Maryland had been for the most part what one might call Old School.
Categories and functors were mentioned, of necessity, in the algebraic topology course I took, but even there they weren't really prominent. The graduate algebra course was now taught out of Lang, and since I was going to have to take the comprehensive exams at UCSD, that made it pretty clear that I needed to learn Lang's approach.
And furthermore, I found myself giving quite a bit of help to my friends among the other graduate students, and that helped me learn the new approaches as well.
And I found that I, like most of the younger mathematicians I met, absolutely loved category theory. On the other hand, I was constantly encountering older mathematicians, especially around the time I first got my Ph. They told me that they could prove things just as easily without it, and a lot of it seemed to them like just a matter of stating fairly simple things in a very complicated way.
And I would just shrug. What they said seemed to be true, but I still really liked category theory. And gradually I came to think that the appropriate rejoinder to these complaints was that category theory is not primarily a method for proving theorems, but rather a framework for structuring and organizing the knowledge one has.
Well, I have to admit that the course on topos theory I took from John Gray during the year I was at the University of Illinois did seem like a whole lot of stating simple things in a complicated way.
But category theory was essential to the work I did on torsion free abelian groups. Or at least it seemed that way to me.Summary: MLA (Modern Language Association) style is most commonly used to write papers and cite sources within the liberal arts and humanities.
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