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This is not exactly a question regarding a hint to solve an exercise of a list or an exam, but a question involving the possible utility of which is my favourite Mathematics branch: group theory. As far as I have heard, group theory has not as much applications as other branches such as Analysis, ODE and ODPs, or geometry. However, there is no doubt that there is a fundamental connection between abstract algebra and some parts of Analysis. Indeed, the largest proof in Mathematics is the one of the Theorem of Classification of Finite Simple Groups, here is a link: Classification of finite simple groups; which was completed in 2004 and supposed between 10000 and 20000 pages of mathematical skills projected in a bunch of documents.
Which applications have abstract algebra in general (not only group theory, but also ring theory, fields, modules, etc) in the professional world (without including teaching or research)? Is it possible to apply pure abstract algebra in a job, or is this just a drem?
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Daniel C.
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2What is "the professional world"? I am a professor, which by definition is a professional: I am paid for my work. So you probably mean something a little narrower than "you are paid to do it." – Arturo Magidin Oct 31 '23 at 19:57
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Exactly, I should have specified any job which is not directly related to teaching. This is, private corporations for example – Daniel C. Oct 31 '23 at 19:59
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1So then say that, not "in the professional world". And are you also excluding research? Not every job involving research involves teaching. And just so you know: the largest employer of mathematicians in the United States is the National Security Agency, where abstract algebra is certainly used. – Arturo Magidin Oct 31 '23 at 20:02
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3Crystallographers use groups to identify molecules by examining the symmetries observed when scatting light off them. Invertible affine transformations form a group and many geometric programming tasks involve affine transformations. – CyclotomicField Oct 31 '23 at 20:02
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3I expect the poster is looking for "real world applications" :) – Derek Holt Oct 31 '23 at 20:09
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Exactly, that is what I was looking for – Daniel C. Oct 31 '23 at 20:11
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2Again: if that is what you want, then you should say so in the post, not in the comments. Use that "Edit" button. And did you search the site? Googling "applications of Group Theory" yielded quite a bit, including this. – Arturo Magidin Oct 31 '23 at 20:14
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4Crystallography. Cryptography. Quantum computing. Quantum mechanics. Knowing you can't have a mindless method for flipping your mattress that doesn't remember what you did the last time.... – Arturo Magidin Oct 31 '23 at 20:17
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2See the answers and comments to Applications of abstract algebra outside of mathematics and suitable textbook AND Are there real world applications of finite group theory? AND Do groups, rings and fields have practical applications in CS? If so, what are some? and these others. – Dave L. Renfro Oct 31 '23 at 20:50