Group theory applications along with a solved example

Question

As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and prove problems related to Group theory.

But when comes to applications, I don't know where to start.

I surfed the net, and I can get these links....

http://www.math.uconn.edu/~kconrad/math216/whygroups.html

http://en.wikiversity.org/wiki/Topic:Group_theory

http://ezinearticles.com/?Why-Study-Math?---Group-Theory-and-Subparticle-Physics&id=1456420

Those explanations are really good. But the real problem I face is, all applications are of theoretical explanations, without a solved example which a beginner like me can understand.

When I went to Wikipedia, I learned about the solution of the Rubik's cube in at most 20 steps posted in http://cube20.org/

What I can understand?

Turning a cube upside down, it will still take the same number of moves to solve.(Symmetrical property).

Where I need assistance?

An example of showing how this symmetrical property of group theory works here.

So, if someone could give an example of how group theory is applied (in this or some other instance) it will be useful to me....

Answer 1

One can analyse Rubik's cube using Gap. See here. I am unsure if this is what you are looking for or not, but it starts by showing how to "see" your cube as a permutation group, and then analyses the permutations.

(Too see your cube as a permutation group, write numbers on all the cubies, apart from the $6$ centre cubies, and see what rotating each of the $6$ faces once does to these numbers.)

Answer 2

As we seem to have gotten started with Rubik's cube let me add a few ways some basic properties of permutation groups can be used to solve that puzzle. The goal is not to find the shortest solution, any solution will do.

1. You may have learned that the alternating group can be generated with 3-cycles. So if we can find sequences of moves that produces 3-cycles on corner pieces and other sequences of moves permuting edge pieces in a 3-cycle, we can get all the tiny cubes into the correct places (but possibly first in wrong positions or should I call them orientations?)
2. An efficient way of producing the necessary 3-cycles is as commutators. A general scheme for that is the following. Assume that we have written the set $X=A\cup B$ as a disjoint union of two proper subsets. Assume that a permutation $\alpha\in S(X)$ only moves the elements of $A$ and keeps all the elements of $B$ fixed. Assume further that $x\in A$ and another permutation $\beta\in S(X)$ keeps all the elements of $A\setminus\{x\}$ fixed. Then it is easy to see that the commutator $\alpha\beta\alpha^{-1}\beta^{-1}$ is a 3-cycle moving only the elements $\{x,\alpha(x),\beta(x)\}$.
3. You can easily produce such 3-cycles on Rubik's cube, because finding suitable permutations $\alpha$ and $\beta$ is easy. For images of such sequences of moves, please visit my page. The surrounding text gives more details, but it is in Finnish only, so you may have a hard time deciphering most of it (@-ping me?). In all those commutators the set $A$ is the upper one third of the cube, and the set $B$ consists of the lower two thirds. The permutation $\alpha$ is thus simply one quarter turn of the top layer. The permutation $\beta$ OTOH varies according to our needs.
4. Similar thinking with commutators allows us to design short sequences of moves that turn two adjacent corner pieces or two adjacent edge pieces. These are also shown on my page.
5. The depicted sequences operate on pieces that are in the 'front'. If we need some other 3-cycle, we can simply first do a few moves with the effect of bringing the desired 3, say, corner pieces into the position of the shown 'standard' 3-cycle on corners. We can then do the 3-cycle, and then as the last step cancel the first couple of moves in reverse order. IOW we produced another 3-cycle by conjugating a given one. Sound familiar?
6. Similarly we can rotate pairs of pieces by a conjugate of the shown sequence, but usually it is easier to just look at the cube from a different direction :-)

Have fun!

Hopefully you noticed that no deep group theory was really used. The upshot is that the language of (permutation) groups allows us to see structure in the chaos of the Cube. Also it allows us to split the problem into small tractable parts.

Don't get me started on coding theory. I tend to view problems in telecommunications as problems of applied algebra (finite fields and their characters, related character sums, vector spaces over them, finite groups and suitable representations, orders in division algebras). So I would lapse into public solipsism. If your only tool is a hammer ...

Answer 3

i think Group Theory is of use in Chemistry and Physics to calculate "Energy Levels" in atoms; you can look it up in the Wikipedia. The group calculations make non linear calculations easier and may have application in the design of drugs.. Lie Groups in Physics come to mind.