What are some real life applications of group theory?

Question

What are some real life applications of group theory?

Answer 1

rubik's cube

Answer 2

Group theory is important in physics, see link



http://sigfpe.blogspot.com/2007/11/whats-all-this-e8-stuff-about-then-part.html



in other problems such as "marriage matchings"

Answer 3

On Open University I saw a program about how a mechanical mail sorter used group theory. It flipped or turned the letters so that the address was facing the right way.

They had another show where they used group theory to count the number of ways to paint a cube with three different colors. Each side completely colored with one color. I don't know practical that is. But one could use it for inventing a game.



ksoileau, can you give us some references, please?



ADDED COMMENTS:

If you are standing, facing north and rotate forward 90 degrees about a left-right axis and then rotate clockwise 90 degrees about a head-to-foot axis, you will be lying down on your left side and facing east. However, if you first turn about the head-to-foot axis and then about the left-right axis, you will lying face down with your head pointing east.



Going to bed is not abelian.



There are probably many everyday activities to which one could apply group theory; addition of integers is a group. But we can solve these common activities without group theory.



The power of group theory is for some conceptually difficult problems.



My difficulty is realizing that group theory is applicable to a problem.

Answer 4

It's useful in error-correcting codes, which makes real-life information transmission more accurate.



It's also useful in particle physics and string theory. Particle physics already has real-life application, and string theory may well do so in the future.

Answer 5

what is a group

http://members.tripod.com/~dogschool/



Now for each law you associate two individual belong to human lets say.

Human is the big set we consider.

for law either closure, associative is a semi group proved



Now The Identity -- is the Vacuum itself



Now Inverse Human is no human then it is all the vacuum



Now coming to your concept of abelian group



if any group that satisfies the commutative with the above four then it is an abelian

If not abelian we have the solution above

if abelian you have the meaning associated to what you want and you have achieved fulfillment.

just the understanding, the sharing , the caring , the love and all the glorious period you walk through for enlightenment



In my view I cite this example for nothing is beyond humanity and its very true existence.



Every law of maths is true life

its only that we see it in the form equations and abstractions, but it is more than science



If it is still unclear please let me know











http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Development_group_theory.html





http://www.amazon.com/Group-Theory-W-R-Scott/dp/0486653773 is truly a very good book





If u read modern algebra by Seth and Sharma and any Indian author you yourself will be able to correlate to real life situations too

Answer 6

Group Theory Applications

Answer 7

This Site Might Help You.



RE:

What are some real life applications of group theory?

What does it matter in real life if a group is abelian or not?



NOTE that I'm not trying to be demeaning, I honestly want to know.

Answer 8

I'll answer your specific question first:



"What does it matter in real life if a group is abelian or not?"



The group SO(3), the group of rotations in Euclidean 3-space, is non-abelian. This directly reflects a pretty ordinary aspect of our world. SO(3) has a variety of applications in physics and computer graphics. For example, SO(3) can be represented by the set of unit quaternions, which permit rotations to be stored in memory and combined together in an efficient manner in applications to computer graphics.



Furthermore, GL(3), the group of 3x3 matrices under multiplication, corresponds to the space of linear transformations of R^3 into itself given the canonical vector space structure, and this group is of course non-abelian. This is a generalization of the previous example of the non-commutativity of SO(3) since all rotations are linear transformations.



The representation theory (through Lie algebras) of non-abelian groups is of tremendous importance to modern physics, especially quantum mechanics. Indeed, the elements of the Lie algebra for the rotation group in quantum mechanics correspond to angular momenta.



As for the general question of applications of group theory, I've already supplied one: quantum mechanics. The fact that some polynomials of degree n > 4 can't be solved in radicals can be shown to be a consequence of the fact that the symmetric group S_n contains a simple, non-cyclic, normal subgroup. This is a great achievement of Galois theory and is known as the Abel-Ruffini theorem.



In particle physics, symmetry groups help describe the interactions of particles, and in particular the notion of a gauge connection is of utmost importance.



In combinatorics, symmetry groups also emerge in Burnside's theorem, which enables the easy solutions of counting problems which would otherwise be unthinkable.