Use of group theory

Please use google before asking a question here.

http://en.wikipedia.org/wiki/Group_theory#Applications_of_group_theory

 

The thing is that groups are a natural object that show up everywhere in mathematics. And I really mean everywhere. If a structure shows up a lot of time, then it makes sense to give it a name. That doesn't explain why it's a useful structure though.

There are certain operation on groups, such as quotients and products. These operations show up quite naturally in mathematics. But if we didn't have an abstract notion of a group, then we had to define all of these operations over and over again. Now we can just say: take the quotient of groups. Everybody will know perfectly what you mean. Without groups, you will have to construct to quotient again.

Furthermore, groups tend to be structures that are reasonably well known to mathematicians. Of course there are many unsolved problems about groups (some of which relate to string theory and fourier analysis!). But overall, a mathematician will feel comfortable when dealing with a group. Now, many hard mathematical problems can be simplified by translating the problem into a problem of groups. And then we can solve the problem by using our knowledge of groups. This is done in (for example) algebraic topology.

You might want to check out the history of group theory. It might be more clear why groups were defined the way they were.

 

There are different aspects of group theory. The Sylow theorem based study of finite groups is not the most useful (outside of finite group theory).

With my limited perspective on algebra, I think they teach it this way because it introduces students about abstract mathematics. Also, it prepares you for Galois theory, which is glorious to algebraists. Personally, I think group representations and the classical Lie groups would have been more useful than group presentations and the Sylow theorems. But perhaps that is too meaty a course and not really comparable to the standard introduction to group theory.

 

For practical examples, you have both concrete applications, such as applications to physics, and abstract applications, such as the fundamental group of a topological space. In the later case, one case see that groups show up as mathematical structures in all kinds of places, so it's worth while to study them by themselves to economize our understanding of mathematics. Abelian groups are also part of one way to define vector spaces. Groups are also a simplified example of a Field, which the real numbers are an (obviously) important example. One last example: groups are used in some results of algebraic number theory. I hope I'm painting a picture where groups show up in lots of places in mathematics, so the particular choice of definition is a useful one.

For more practical examples, you can look to physics. Any kind of symmetry can be modeled using group theory (if you like, you can use that feature to motivate the definition of a group). Lorentz transformations, which relate two different reference frames in special relativity, form a group. Also other symmetries show up, called "gauge symmetries", show up in quantum field theory, and these also form groups. In general relativity, the R coordinate in the Schwarzchild metric is defined using orbits of a group. By studying group representation theory (closely related to group theory), one can derive the quantized "spectrum" of angular momentum in quantum mechanics.

Did you study the symmetry group of any object? If not, perhaps the best thing you could do is find some object with symmetry, and consider it's "symmetry transformations". Convince yourself that the set of symmetry transformations are a group: is there an identity transformation? is there an inverse transformation? is the action of two symmetries also a symmetry? are symmetries associative?

The answer to all those questions are yes, but you should really convince yourself that its true by physically holding something that is symmetric and rotating it accordingly.

 

The first time I came across group theory was in chemistry. Take the molecule Fe(CO)₅. This compound has two forms: the square pyramid formation and the formation that is on the page (I forgot what that formation is called).

Anyway, the question is can we detect which form our sample is? Here is where group theory comes in. The square pyramid has symmetry group C₄ while the other form has symmetry group D₃. This means when we place the samples in an IR spectrometer, we will get different readings.

In order to predict which produces what signal we use representation theory of groups. This means we write the group elements as matrices. But there are always "good" ways to do this. We thus are able to reduce our representation in to a linear combination of these "good" (irreducible) representations. So we might end up with 2A+B. What this means is on the IR reading, we should see a single peak and a separate double peak.

 

Group theory is extensively utilized in many aspects of x-ray crystallography, inorganic chemistry, protein analysis, organic chemistry, chemical physics, physical chemistry, theoretical chemistry, computational analysis, and so forth (and this list is not exhaustive).

If any molecular or atomic system you are examining has any sort of symmetry, group theory is very useful in its analysis and computation.

 

Do you think that the idea of a 1-to-1 function from a set onto itself is a useful concept? (If not, then perhaps you should give up mathematics!) Such functions have inverse functions that are 1-to-1. If you have a set of such functions (including the identity function) and their inverses then all ways of composing (i.e f(g(x)) ) these functions "generates" a group under the operation of composition. In fact, all groups can be regarded as being this type of group.

As to the uses of group theory, you don't see any dramatic applications of group theory until you get to very advanced courses. In elementary courses the knowledge of groups gives a concise way to stating things. For example, you can say that the vectors in a vectors space form an abelian group under vector addition. You know from group theory that each element in a group has a unique inverse. So you don't have say each vector V has an unique vector that is its additive inverse since that's a known result from group theory.