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Application field can vary from biology\biochemistry, to computer science\coding theory, the more unexpected a connection to a field, the better. And paper preferably should be not very large one.

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up vote 2 down vote accepted

Tom Fiore and his coauthors have used group theory to study various pieces of music. You might enjoy the article "Musical actions of dihedral groups ( ), or take a look at Tom's music page:

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Here is a paper about applications of group theory to symmetric gates.

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See Polya's book for an application of Burnside's lemma to chemistry.

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There are many applications of group theory to chemistry, in particular to fullerenes. If you do a "google search" with the strings: +chemistry +"group theory" or +"group theory" +fullerenes you will find many papers that deal with these topics.

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