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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.
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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.A guide to help users create citations using APA (American Psychological Association) style, 6th edition.
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