Applications of Abstract Algebra?

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What are some applications of abstarct algebra? I have to write a paper and present on a application of abstract algebra and am looking for topic ideas.
 
Cryptography is a good application of it. Also, a lot of quantum mechanics stuff is derived from abstract algebra.
 
Is there anything I could do with combinatorics? Would this involve permutations possibly?
 
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You could do something like an enumeration of the allowed configurations of the Rubik's cube. The problem is actually quite nice, in that the solution isn't overly complicated and the answer, which characterizes allowed configurations in terms of the signatures of various permutations of sub-cubes, is fairly easy to state.

If you're looking specifically for interesting applications to combinatorics, you could also try starting with Burnside's Lemma or the Redfield-Polya theorem, both of which are truly beautiful (and powerful) results in enumerative combinatorics that tell you how to count states of things modulo some symmetry group. (For example, using R-P, you can answer questions like "In how many distinguishable ways can I paint the faces of an icosahedron using six different colors, such that exactly 4 of the faces are red?")
 
You could do something like an enumeration of the allowed configurations of the Rubik's cube. The problem is actually quite nice, in that the solution isn't overly complicated and the answer, which characterizes allowed configurations in terms of the signatures of various permutations of sub-cubes, is fairly easy to state.

If you're looking specifically for interesting applications to combinatorics, you could also try starting with Burnside's Lemma or the Redfield-Polya theorem, both of which are truly beautiful (and powerful) results in enumerative combinatorics that tell you how to count states of things modulo some symmetry group. (For example, using R-P, you can answer questions like "In how many distinguishable ways can I paint the faces of an icosahedron using six different colors, such that exactly 4 of the faces are red?")
Ok, I'm doing campanology or the study of bell ringing. I got a lot of good nfo, but is there any good exercises to go along with my paper possibly?
 
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First discrete abstract algebra was developed while trying to understand polynomials and their roots. Lie generalized it as continuous groups to solve differential equations. All though now you can find and use differential equation symmetries with little group theory knowledge (but much more linear algebra) the method was still developed using abstract algebra.

Second symmetry groups are used in studying and classifying symmetric objects like viruses and crystals.

Also there are topics like quantum groups and representation theory which are also abstract algebra (but I do not know their details).

Finite groups are used in coding and information theory.

Rings and fields are used in cryptology.

That is all I can remember for now.
 

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What are some applications of abstarct algebra? I have to write a paper and present on a application of abstract algebra and am looking for topic ideas.
Error control coding (aka error correction coding) uses a fair amount of field theory. This is a crucial part of many digital communication systems (cell phones, satellite TV, etc.) and also digital storage formats (CDs and DVDs).
 

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