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?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?")
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).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.