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The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra.

What are some applications of abstract algebra outside of mathematics? These could be included either in a traditional abstract algebra course or in an applied algebra course. Which abstract algebra textbooks have particularly good support for applications or have a focus on applied algebra?

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    $\begingroup$ You might look up “point groups” in physical chemistry. $\endgroup$
    – Steve
    Oct 9 '20 at 13:53
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    $\begingroup$ I imagine there will be several answers with specifics, but if not and you're looking for something to mention (or possibly even include) when teaching, you can simply google something like "group theory" (probably holds much more promise than anything else I can think of right now) along with various scientific fields/topics, such as this. Off the top of my head I know that most physical chemistry texts and quantum chemistry texts have chapters on group theory. (While I was writing my comment @Steve gave this example.) $\endgroup$
    – Dave L Renfro
    Oct 9 '20 at 13:57
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    $\begingroup$ The calculations in algebraic topology boil down to algebra after a while. I believe modern cutting edge quantum mechanics now involves such topology and hence such algebra. More classically, tesselations and finding shapes which are invariant under transformations leads to designs which are more or less applied abstract algebra. Of course such designs may precede the algebra, but they are better understood in the light of the algebra. Then, representation theory in physics is applied abstract algebra, all the "quantum numbers" are born of representation theory which is abstract algebra... $\endgroup$
    – James S. Cook
    Oct 9 '20 at 15:38
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    $\begingroup$ Feel free to post comments like the above as answers. $\endgroup$
    – J W
    Oct 9 '20 at 16:54
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    $\begingroup$ My answer to the linked question listed applications in telecommunication engineering. Coding theory and cryptography in particular are full of applied algebra. But, I cannot recommend a textbook covering those themes in a book dedicated to algebra. Sure, some of the applications are widely known, but not many authors of books on algebra are familiar with too many of them. My coverage (when given a chance) is based on first hand experience rather than textbooks. $\endgroup$
    – Jyrki Lahtonen
    Oct 9 '20 at 20:24
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Thanks in part to Jyrki Lahtonen's remarks on coding theory and cryptography, I have an applied algebra book to suggest:

Algebra for Applications, Arkadii Slinko, 2nd ed., 2020 Springer

The preface to the first edition states:

This book originated from my lecture notes for the one-semester course which I have given many times in The University of Auckland since 1998. The goal of this book is to show the incredible power of algebra and number theory in the real world. It does not advance far in theoretical algebra, theoretical number theory or combinatorics. Instead, we concentrate on concrete objects like groups of points on elliptic curves, polynomial rings and finite fields, study their elementary properties and show their exceptional applicability to various problems in information handling. Among the applications are cryptography, secret sharing, error-correcting, fingerprinting and compression of information.

It does not replace a more traditional course on abstract algebra and indeed at The University of Auckland where Slinko teaches Algebra and Applications, a conventional course, Algebraic Structures, is also taught (see list of undergraduate courses).

For an MAA review of the first edition by Mark Hunacek, see https://www.maa.org/press/maa-reviews/algebra-for-applications, noting that an index has now been added to the second edition.

Whether Slinko's book counts as "a book dedicated to algebra", to borrow Jyrki Lahtonen's phrase, is debatable I suppose. I think Slinko tries to strike a balance between introducing the necessary algebra and giving sufficient space to the applications. I also think that in general, the line is blurred somewhat between algebra, number theory and applications. For example, some searching has revealed Niederreiter & Winterhof's Applied Number Theory, which after a review of some number theory and algebra, covers cryptography, error-correcting codes, quasi-Monte Carlo methods, pseudorandom numbers and miscellaneous further applications. Another example is Hoffstein, Pipher & Silverman's An Introduction to Mathematical Cryptography, which naturally focuses on cryptography, but introduces the necessary topics from number theory, algebra, probability and information theory to keep the book fairly self-contained. The contents of these books and Slinko's book clearly overlap to some extent.

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There are a lot of applications of group theory in chemistry and physics. (I know the chemistry side of it better.)

Point groups of molecules describe their symmetry and thus their spectroscopy (bending modes for IR spectra). See for instance,

https://www.amazon.com/Chemical-Applications-Group-Theory-3rd/dp/0471510947

OR

https://www.youtube.com/watch?v=BjknQc3bQ8o (note the character tables)

OR

https://www.youtube.com/watch?v=pEw0RKCANFs (simple visual)

Space groups are intrinsic to X-ray crystallography, which describes a huge amount of matter (even things you don't think of as crystals, like metals or ceramics or [bulk of] computer chips, are crystalline solids). See for instance

https://it.iucr.org/

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This is a classic article:

Hayes, Brian. Group Theory in the Bedroom, and other Mathematical Diversions. Hill and Wang, 2008. Book link. "Having run out of sheep the other night, I found myself counting the ways to flip a mattress. ..."

           

(The title refers to one article. The entire collection is worthwhile.)

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